Abstract

During this last decade, several remarkable phenomena inherent to the nonlinear propagation of incoherent optical waves have been reported in the literature. This article is aimed at providing a generalized wave turbulence kinetic formulation of random nonlinear waves governed by the nonlinear Schrödinger equation in the presence of a nonlocal or a noninstantaneous nonlinear response function. Depending on the amount of nonlocal (noninstantaneous) nonlinear interaction and the amount of inhomogeneous (nonstationary) statistics of the incoherent wave, different types of kinetic equations are obtained. In the spatial domain, when the incoherent wave exhibits fluctuations that are statistically homogeneous in space, the relevant kinetic equation is the wave turbulence (Hasselmann) kinetic equation. It describes, in particular, the process of optical wave thermalization to thermodynamic equilibrium, which slows down significantly as the interaction becomes highly nonlocal. When the incoherent wave is characterized by inhomogeneous statistical fluctuations, different forms of the Vlasov equation are derived, which depend on the amount of nonlocality in the system. This Vlasov approach describes, in particular, the processes of incoherent modulational instability and the formation of localized incoherent soliton structures. In the temporal domain, the noninstantaneous nonlinear response function is constrained by the causality condition. It turns out that the relevant kinetic equation has a form analogous to the weak Langmuir turbulence equation, which describes, in particular, the formation of nonlocalized spectral incoherent solitons. In the regime of a highly noninstantaneous nonlinear response and a stationary statistics of the incoherent wave, the weak Langmuir turbulence equation reduces to the Korteweg–de Vries equation. Conversely, in the regime of a highly noninstantaneous response in the presence of a nonstationary statistics, we derive a long-range Vlasov-like kinetic equation in the temporal domain, whose self-consistent potential is constrained by the causality condition. From a broader perspective, this work indicates that the wave turbulence theory may constitute the appropriate theoretical framework to formulate statistical nonlinear optics.

© 2012 Optical Society of America

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  1. R. W. Boyd, Nonlinear Optics (Academic Press, 2008).
  2. Y. S. Kivshar and G. P. Agrawal, Optical Solitons : From Fibers to Photonic Crystals (Academic Press, 2003).
  3. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  4. M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
    [CrossRef]
  5. M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature (London) 387, 880–883 (1997).
    [CrossRef]
  6. D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media” Phys. Rev. Lett. 78, 646–649 (1997).
    [CrossRef]
  7. M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
    [CrossRef]
  8. O. Bang, D. Edmundson, and W. Krolikowski, “Collapse of incoherent light beams in inertial bulk Kerr media,” Phys. Rev. Lett. 83, 5479–5482 (1999).
    [CrossRef]
  9. W. Krolikowski, O. Bang, and J. Wyller, “Nonlocal incoherent solitons,” Phys. Rev. E 70, 036617 (2004).
    [CrossRef]
  10. M. Peccianti and G. Assanto, “Incoherent spatial solitary waves in nematic liquid crystals,” Opt. Lett. 26, 1791–1793 (2001).
    [CrossRef]
  11. M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000).
    [CrossRef]
  12. D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
    [CrossRef]
  13. A. Sauter, S. Pitois, G. Millot, and A. Picozzi, “Incoherent modulation instability in instantaneous nonlinear Kerr media,” Opt. Lett. 30, 2143–2145 (2005).
    [CrossRef]
  14. D. N. Christodoulides, T. H. Coskun, M. Mitchell, Z. Chen, and M. Segev, “Theory of incoherent dark solitons,” Phys. Rev. Lett. 80, 5113–5116 (1998).
    [CrossRef]
  15. Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998).
    [CrossRef]
  16. H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides, “Random-phase solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 92, 223901 (2004).
    [CrossRef]
  17. O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Observation of random-phase lattice solitons,” Nature (London) 433, 500–503(2005).
    [CrossRef]
  18. G. A. Pasmanik, “Self-interaction of incoherent light beams,” Sov. Phys. JETP 39, 234–238 (1974).
  19. M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
    [CrossRef]
  20. D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media,” Phys. Rev. Lett. 78, 646–649 (1997).
    [CrossRef]
  21. B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002).
    [CrossRef]
  22. T. Hansson, D. Anderson, M. Lisak, V. E. Semenov, and U. Osterberg, “Propagation of partially coherent light beams with parabolic intensity distribution in noninstantaneous nonlinear Kerr media,” J. Opt. Soc. Am. B 25, 1780–1785 (2008).
    [CrossRef]
  23. D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media,” Phys. Rev. E 63, 035601 (2001).
    [CrossRef]
  24. M. Lisak, L. Helczynski, and D. Anderson, “Relation between different formalisms describing partially incoherent wave propagation in nonlinear optical media,” Opt. Commun. 220, 321–323 (2003).
    [CrossRef]
  25. N. Akhmediev, W. Krolikowski, and A. W. Snyder, “Partially voherent solitons of variable shape,” Phys. Rev. Lett. 81, 4632–4635 (1998).
    [CrossRef]
  26. T. Hansson, M. Lisak, and D. Anderson, “Integrability and conservation laws for the nonlinear evolution equations of partially coherent waves in noninstantaneous Kerr media,” Phys. Rev. Lett. 108, 063901 (2012).
    [CrossRef]
  27. I. B. Bernstein, J. M. Green, and M. D. Kruskal, “Exact nonlinear plasma oscillations,” Phys. Rev. 108, 546–550 (1957).
    [CrossRef]
  28. A. Hasegawa, “Dynamics of an ensemble of plane waves in nonlinear dispersive media,” Phys. Fluids 18, 77–79 (1975).
    [CrossRef]
  29. A. Hasegawa, “Envelope soliton of random phase waves,” Phys. Fluids 20, 2155–2156 (1977).
    [CrossRef]
  30. D. V. Dylov and J. W. Fleischer, “Observation of all-optical bump-on-tail instability,” Phys. Rev. Lett. 100, 103903 (2008).
    [CrossRef]
  31. J. Garnier, J.-P. Ayanides, and O. Morice, “Propagation of partially coherent light with the Maxwell–Debye equation,” J. Opt. Soc. Am. B 20, 1409–1417 (2003).
    [CrossRef]
  32. J. Garnier and A. Picozzi, “Unified kinetic formulation of incoherent waves propagating in nonlinear media with noninstantaneous response,” Phys. Rev. A 81, 033831 (2010).
    [CrossRef]
  33. O. Cohen, H. Buljan, T. Schwartz, J. Fleischer, and M. Segev, “Incoherent solitons in instantaneous nonlocal nonlinear media,” Phys. Rev. E 73, 015601 (2006).
    [CrossRef]
  34. C. Rotschild, T. Schwartz, O. Cohen, and M. Segev, “Incoherent spatial solitons in effectively-instantaneous nonlocal nonlinear media,” Nat. Photon. 2, 371–376 (2008).
    [CrossRef]
  35. A. Picozzi and J. Garnier, “Incoherent soliton turbulence in nonlocal nonlinear media,” Phys. Rev. Lett. 107, 233901(2011).
    [CrossRef]
  36. A. Picozzi and M. Haelterman, “Parametric three-wave soliton generated from incoherent light,” Phys. Rev. Lett. 86, 2010–2013 (2001).
    [CrossRef]
  37. A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, “Incoherent solitons in instantaneous response nonlinear media,” Phys. Rev. Lett. 92, 143906 (2004).
    [CrossRef]
  38. M. Wu, P. Krivosik, B. A. Kalinikos, and C. E. Patton, “Random generation of coherent solitary waves from incoherent waves,” Phys. Rev. Lett. 96, 227202 (2006).
    [CrossRef]
  39. T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, “The physics of dipolar bosonic quantum gases,” Rep. Prog. Phys. 72, 126401 (2009).
    [CrossRef]
  40. S. Skupin, M. Saffman, and W. Krolikowski, “Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor,” Phys. Rev. Lett. 98, 263902 (2007).
    [CrossRef]
  41. C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003).
    [CrossRef]
  42. C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
    [CrossRef]
  43. M. Segev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
    [CrossRef]
  44. N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, “Shocks in nonlocal media,” Phys. Rev. Lett. 99, 043903 (2007).
    [CrossRef]
  45. C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
    [CrossRef]
  46. A. G. Litvak and A. M. Sergeev, “One dimensional collapse of plasma waves,” JETP Lett. 27, 517–520 (1978).
  47. J. Wyller, W. Krolikowski, O. Bang, and J. J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66, 066615 (2002).
    [CrossRef]
  48. O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619 (2002).
    [CrossRef]
  49. S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, “Stability of two-dimensional spatial solitons in nonlocal nonlinear media,” Phys. Rev. E 73, 066603 (2006).
    [CrossRef]
  50. A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, “Observation of attraction between dark solitons,” Phys. Rev. Lett. 96, 043901 (2006).
    [CrossRef]
  51. W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
    [CrossRef]
  52. A. Snyder and D. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
    [CrossRef]
  53. V. E. Zakharov, A. N. Pushkarev, V. F. Shvets, and V. V. Yan’kov, “Soliton turbulence,” JETP Lett. 48, 83–87 (1988).
  54. R. Jordan and C. Josserand, “Self-organization in nonlinear wave turbulence,” Phys. Rev. E 61, 1527–1539 (2000).
    [CrossRef]
  55. B. Rumpf and A. C. Newell, “Coherent structures and entropy in constrained, modulationally unstable, nonintegrable systems,” Phys. Rev. Lett. 87, 054102 (2001).
    [CrossRef]
  56. B. Rumpf and A. C. Newell, “Localization and coherence in nonintegrable systems,” Physica D 184, 162–191 (2003).
    [CrossRef]
  57. K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374, 3585–3589 (2010).
    [CrossRef]
  58. V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, “Hamiltonian approach to the description of non-linear plasma phenomena,” Phys. Rep. 129, 285–366 (1985).
    [CrossRef]
  59. M. Onorato, A. Osborne, R. Fedele, and M. Serio, “Landau damping and coherent structures in narrow-banded 1+1 deep water gravity waves,” Phys. Rev. E 67, 046305 (2003).
    [CrossRef]
  60. A. Campa, T. Dauxois, and S. Ruffo, “Statistical mechanics and dynamics of solvable models with long-range interactions,” Phys. Rep. 480, 57–159 (2009).
    [CrossRef]
  61. A. Picozzi, S. Pitois, and G. Millot, “Spectral incoherent solitons: a localized soliton behavior in the frequency domain,” Phys. Rev. Lett. 101, 093901 (2008).
    [CrossRef]
  62. C. Michel, B. Kibler, and A. Picozzi, “Discrete spectral incoherent solitons in nonlinear media with noninstantaneous response,” Phys. Rev. A 83, 023806 (2011).
    [CrossRef]
  63. B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in photonic crystal fiber,” Phys. Rev. E 84, 066605 (2011).
    [CrossRef]
  64. S. L. Musher, A. M. Rubenchik, and V. E. Zakharov, “Weak Langmuir turbulence,” Phys. Rep. 252, 177–274 (1995).
    [CrossRef]
  65. Y. B. Zel’dovich, E. V. Levich, and R. A. Syunyaev, “Stimulated Compton interaction between Maxwellian electrons and spectrally narrow radiation,” Sov. Phys. JETP 35, 733–740 (1972).
  66. C. Montes, J. Peyraud, and M. Hénon, “One-dimensional boson soliton collisions,” Phys. Fluids 22, 176–182 (1979).
    [CrossRef]
  67. V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, “Weak Langmuir turbulence of an isothermal plasma,” Sov. Phys. JETP 42, 80–86 (1975).
  68. C. Montes, “Photon soliton and fine structure due to nonlinear Compton scattering,” Phys. Rev. A 20, 1081–1095 (1979).
    [CrossRef]
  69. V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I (Springer, 1992).
  70. V. E. Zakharov, F. Dias, and A. Pushkarev, “One-dimensional wave turbulence,” Phys. Rep. 398, 1–65 (2004).
    [CrossRef]
  71. A. C. Newell, “The closure problem in a system of random gravity waves,” Rev. Geophys. 6, 1–31 (1968).
    [CrossRef]
  72. A. C. Newell, S. Nazarenko, and L. Biven, “Wave turbulence and intermittency,” Physica D 152, 520–550 (2001).
    [CrossRef]
  73. A. C. Newell and B. Rumpf, “Wave turbulence,” Ann. Rev. Fluids Mech. 43, 59–78 (2001).
  74. S. Nazarenko, Wave Turbulence, Lectures Notes in Physics825 (Springer, 2011).
  75. K. Hasselmann, “On the non-linear energy transfer in a gravity-wave spectrum. Part 1. General theory,” J. Fluid Mech. 12, 481–500 (1962).
    [CrossRef]
  76. K. Hasselmann, “On the non-linear energy transfer in a gravity-wave spectrum. Part 2. Conservation theorems; wave-particle analogy; irreversibility,” J. Fluid Mech. 15, 273–281 (1963).
    [CrossRef]
  77. A. Picozzi and P. Aschieri, “Influence of dispersion on the resonant interaction between three incoherent waves,” Phys. Rev. E 72, 046606 (2005).
    [CrossRef]
  78. S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity locking of incoherent nonlinear wave packets,” Phys. Rev. Lett. 97, 033902 (2006).
    [CrossRef]
  79. A. Picozzi, “Spontaneous polarization induced by natural thermalization of incoherent light,” Opt. Express 16, 17171–17185 (2008).
    [CrossRef]
  80. A. Picozzi and S. Rica, “Coherence absorption and condensation induced by thermalization of incoherent nonlinear fields,” Europhys. Lett. 84, 34004 (2008).
    [CrossRef]
  81. P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, “Anomalous thermalization of nonlinear wave systems,” Phys. Rev. Lett. 104, 054101 (2010).
    [CrossRef]
  82. C. Michel, P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, “Influence of third-order dispersion on the propagation of incoherent light in optical fibers,” Opt. Lett. 35, 2367–2369 (2010).
    [CrossRef]
  83. D. B. S. Soh, J. P. Koplow, S. W. Moore, K. L. Schroder, and W. L. Hsu, “The effect of dispersion on spectral broadening of incoherent continuous-wave light in optical fibers,” Opt. Express 18, 22393–22405 (2010).
    [CrossRef]
  84. P. Suret, A. Picozzi, and S. Randoux, “Wave turbulence in integrable systems: nonlinear propagation of incoherent optical waves in single-mode fibers,” Opt. Express 19, 17852–17863 (2011).
    [CrossRef]
  85. C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005).
    [CrossRef]
  86. G. Düring, A. Picozzi, and S. Rica, “Breakdown of weak-turbulence and nonlinear wave condensation,” Physica D 238, 1524–1549 (2009).
    [CrossRef]
  87. M. J. Davis, S. A. Morgan, and K. Burnett, “Simulations of Bose fields at finite temperature,” Phys. Rev. Lett. 87, 160402 (2001).
    [CrossRef]
  88. P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi, “Condensation and thermalization of classsical optical waves in a waveguide,” Phys. Rev. A 83, 033838 (2011).
    [CrossRef]
  89. B. Barviau, B. Kibler, S. Coen, and A. Picozzi, “Towards a thermodynamic description of supercontinuum generation,” Opt. Lett. 33, 2833–2835 (2008).
    [CrossRef]
  90. B. Barviau, B. Kibler, and A. Picozzi, “Wave-turbulence approach of supercontinuum generation: influence of self-steepening and higher-order dispersion,” Phys. Rev. A 79, 063840 (2009).
    [CrossRef]
  91. B. Barviau, B. Kibler, A. Kudlinski, A. Mussot, G. Millot, and A. Picozzi, “Experimental signature of optical wave thermalization through supercontinuum generation in photonic crystal fiber,” Opt. Express 17, 7392–7406 (2009).
    [CrossRef]
  92. S. Lagrange, H. R. Jauslin, and A. Picozzi, “Thermalization of the dispersive three-wave interaction,” Europhys. Lett. 79, 64001 (2007).
    [CrossRef]
  93. U. Bortolozzo, J. Laurie, S. Nazarenko, and S. Residori, “Optical wave turbulence and the condensation of light,” J. Opt. Soc. Am. B 26, 2280–2284 (2009).
    [CrossRef]
  94. Y. Silberberg, Y. Lahini, E. Small, and R. Morandotti, “Universal correlations in a nonlinear periodic 1D system,” Phys. Rev. Lett. 102, 233904 (2009).
    [CrossRef]
  95. C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, “Highly noninstantaneous solitons in liquid-core photonic crystal fibers,” Phys. Rev. Lett. 105, 263902 (2010).
    [CrossRef]
  96. A. Picozzi and M. Haelterman, “Condensation in Hamiltonian parametric wave interaction,” Phys. Rev. Lett. 92, 103901 (2004).
    [CrossRef]
  97. C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, “Condensation in disordered lasers: theory, 3D+1 simulations, and experiments,” Phys. Rev. Lett. 101, 143901 (2008).
    [CrossRef]
  98. R. Weill, B. Fischer, and O. Gat, “Light-mode condensation in actively-mode-locked lasers,” Phys. Rev. Lett. 104, 173901 (2010).
    [CrossRef]
  99. R. Weill, B. Levit, A. Bekker, O. Gat, and B. Fischer, “Laser light condensate: experimental demonstration of light-mode condensation in actively mode locked laser,” Opt. Express 18, 16520–16525 (2010).
    [CrossRef]
  100. J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468, 545–548 (2010).
    [CrossRef]
  101. C. Barsi, W. Wan, and J. W. Fleischer, “Imaging through nonlinear media via digital holography,” Nat. Photon. 3, 211–215 (2009).
    [CrossRef]
  102. A. Picozzi, “Entropy and degree of polarization for nonlinear optical waves,” Opt. Lett. 29, 1653–1655 (2004).
    [CrossRef]
  103. E. G. Turitsyna, G. Falkovich, V. K. Mezentsev, and S. K. Turitsyn, “Optical turbulence and spectral condensate in long-fiber lasers,” Phys. Rev. A 80, 031804 (2009).
    [CrossRef]
  104. S. Babin, D. Churkin, A. Ismagulov, S. Kablukov, and E. Podivilov, “Four-wave-mixing-induced turbulent spectral broadening in a long Raman fiber laser,” J. Opt. Soc. Am. B 24, 1729–1738 (2007).
    [CrossRef]
  105. S. A. Babin, V. Karalekas, E. V. Podivilov, V. K. Mezentsev, P. Harper, J. D. Ania-Castanon, and S. K. Turitsyn, “Turbulent broadening of optical spectra in ultralong Raman fiber lasers,” Phys. Rev. A 77, 033803 (2008).
    [CrossRef]
  106. S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castanon, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nat. Photon. 4, 231–235 (2010).
    [CrossRef]
  107. C. Michel, M. Haelterman, P. Suret, S. Randoux, R. Kaiser, and A. Picozzi, “Thermalization and condensation in an incoherently pumped passive optical cavity,” Phys. Rev. A 84, 033848 (2011).
    [CrossRef]
  108. Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury Brown and Twiss interferometry with interacting photons,” Nat. Photon. 4, 721–726 (2010).
    [CrossRef]
  109. B. Kibler, C. Michel, J. Garnier, and A. Picozzi, “Temporal dynamics of incoherent waves in noninstantaneous response nonlinear Kerr media,” Opt. Lett. 37, 2472–2474 (2012).
  110. A. Picozzi and M. Haelterman, “Hidden coherence along space-time trajectories in parametric wave mixing,” Phys. Rev. Lett. 88, 083901 (2002).
    [CrossRef]
  111. A. Piskarskas, V. Pyragaite, and A. Stabinis, “Generation of coherent waves by frequency up-conversion and down-conversion of incoherent light,” Phys. Rev. A 82, 053817 (2010).
    [CrossRef]
  112. A. Stabinis, V. Pyragaite, G. Tamoauskas, and A. Piskarskas, “Spectrum of second-harmonic radiation generated from incoherent light,” Phys. Rev. A 84, 043813 (2011).
    [CrossRef]
  113. G. Strömqvist, V. Pasiskevicius, C. Canalias, P. Aschieri, A. Picozzi, and C. Montes, “Temporal coherence in mirrorless optical parametric oscillators,” J. Opt. Soc. Am. B 29, 1194–1202 (2012).
    [CrossRef]

2012 (3)

2011 (7)

A. Stabinis, V. Pyragaite, G. Tamoauskas, and A. Piskarskas, “Spectrum of second-harmonic radiation generated from incoherent light,” Phys. Rev. A 84, 043813 (2011).
[CrossRef]

P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi, “Condensation and thermalization of classsical optical waves in a waveguide,” Phys. Rev. A 83, 033838 (2011).
[CrossRef]

C. Michel, M. Haelterman, P. Suret, S. Randoux, R. Kaiser, and A. Picozzi, “Thermalization and condensation in an incoherently pumped passive optical cavity,” Phys. Rev. A 84, 033848 (2011).
[CrossRef]

P. Suret, A. Picozzi, and S. Randoux, “Wave turbulence in integrable systems: nonlinear propagation of incoherent optical waves in single-mode fibers,” Opt. Express 19, 17852–17863 (2011).
[CrossRef]

A. Picozzi and J. Garnier, “Incoherent soliton turbulence in nonlocal nonlinear media,” Phys. Rev. Lett. 107, 233901(2011).
[CrossRef]

C. Michel, B. Kibler, and A. Picozzi, “Discrete spectral incoherent solitons in nonlinear media with noninstantaneous response,” Phys. Rev. A 83, 023806 (2011).
[CrossRef]

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in photonic crystal fiber,” Phys. Rev. E 84, 066605 (2011).
[CrossRef]

2010 (12)

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374, 3585–3589 (2010).
[CrossRef]

J. Garnier and A. Picozzi, “Unified kinetic formulation of incoherent waves propagating in nonlinear media with noninstantaneous response,” Phys. Rev. A 81, 033831 (2010).
[CrossRef]

P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, “Anomalous thermalization of nonlinear wave systems,” Phys. Rev. Lett. 104, 054101 (2010).
[CrossRef]

Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury Brown and Twiss interferometry with interacting photons,” Nat. Photon. 4, 721–726 (2010).
[CrossRef]

J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468, 545–548 (2010).
[CrossRef]

R. Weill, B. Fischer, and O. Gat, “Light-mode condensation in actively-mode-locked lasers,” Phys. Rev. Lett. 104, 173901 (2010).
[CrossRef]

C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, “Highly noninstantaneous solitons in liquid-core photonic crystal fibers,” Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castanon, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nat. Photon. 4, 231–235 (2010).
[CrossRef]

A. Piskarskas, V. Pyragaite, and A. Stabinis, “Generation of coherent waves by frequency up-conversion and down-conversion of incoherent light,” Phys. Rev. A 82, 053817 (2010).
[CrossRef]

C. Michel, P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, “Influence of third-order dispersion on the propagation of incoherent light in optical fibers,” Opt. Lett. 35, 2367–2369 (2010).
[CrossRef]

R. Weill, B. Levit, A. Bekker, O. Gat, and B. Fischer, “Laser light condensate: experimental demonstration of light-mode condensation in actively mode locked laser,” Opt. Express 18, 16520–16525 (2010).
[CrossRef]

D. B. S. Soh, J. P. Koplow, S. W. Moore, K. L. Schroder, and W. L. Hsu, “The effect of dispersion on spectral broadening of incoherent continuous-wave light in optical fibers,” Opt. Express 18, 22393–22405 (2010).
[CrossRef]

2009 (10)

B. Barviau, B. Kibler, A. Kudlinski, A. Mussot, G. Millot, and A. Picozzi, “Experimental signature of optical wave thermalization through supercontinuum generation in photonic crystal fiber,” Opt. Express 17, 7392–7406 (2009).
[CrossRef]

U. Bortolozzo, J. Laurie, S. Nazarenko, and S. Residori, “Optical wave turbulence and the condensation of light,” J. Opt. Soc. Am. B 26, 2280–2284 (2009).
[CrossRef]

E. G. Turitsyna, G. Falkovich, V. K. Mezentsev, and S. K. Turitsyn, “Optical turbulence and spectral condensate in long-fiber lasers,” Phys. Rev. A 80, 031804 (2009).
[CrossRef]

B. Barviau, B. Kibler, and A. Picozzi, “Wave-turbulence approach of supercontinuum generation: influence of self-steepening and higher-order dispersion,” Phys. Rev. A 79, 063840 (2009).
[CrossRef]

Y. Silberberg, Y. Lahini, E. Small, and R. Morandotti, “Universal correlations in a nonlinear periodic 1D system,” Phys. Rev. Lett. 102, 233904 (2009).
[CrossRef]

G. Düring, A. Picozzi, and S. Rica, “Breakdown of weak-turbulence and nonlinear wave condensation,” Physica D 238, 1524–1549 (2009).
[CrossRef]

C. Barsi, W. Wan, and J. W. Fleischer, “Imaging through nonlinear media via digital holography,” Nat. Photon. 3, 211–215 (2009).
[CrossRef]

T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, “The physics of dipolar bosonic quantum gases,” Rep. Prog. Phys. 72, 126401 (2009).
[CrossRef]

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef]

A. Campa, T. Dauxois, and S. Ruffo, “Statistical mechanics and dynamics of solvable models with long-range interactions,” Phys. Rep. 480, 57–159 (2009).
[CrossRef]

2008 (9)

A. Picozzi, S. Pitois, and G. Millot, “Spectral incoherent solitons: a localized soliton behavior in the frequency domain,” Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef]

C. Rotschild, T. Schwartz, O. Cohen, and M. Segev, “Incoherent spatial solitons in effectively-instantaneous nonlocal nonlinear media,” Nat. Photon. 2, 371–376 (2008).
[CrossRef]

D. V. Dylov and J. W. Fleischer, “Observation of all-optical bump-on-tail instability,” Phys. Rev. Lett. 100, 103903 (2008).
[CrossRef]

S. A. Babin, V. Karalekas, E. V. Podivilov, V. K. Mezentsev, P. Harper, J. D. Ania-Castanon, and S. K. Turitsyn, “Turbulent broadening of optical spectra in ultralong Raman fiber lasers,” Phys. Rev. A 77, 033803 (2008).
[CrossRef]

A. Picozzi and S. Rica, “Coherence absorption and condensation induced by thermalization of incoherent nonlinear fields,” Europhys. Lett. 84, 34004 (2008).
[CrossRef]

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, “Condensation in disordered lasers: theory, 3D+1 simulations, and experiments,” Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

T. Hansson, D. Anderson, M. Lisak, V. E. Semenov, and U. Osterberg, “Propagation of partially coherent light beams with parabolic intensity distribution in noninstantaneous nonlinear Kerr media,” J. Opt. Soc. Am. B 25, 1780–1785 (2008).
[CrossRef]

A. Picozzi, “Spontaneous polarization induced by natural thermalization of incoherent light,” Opt. Express 16, 17171–17185 (2008).
[CrossRef]

B. Barviau, B. Kibler, S. Coen, and A. Picozzi, “Towards a thermodynamic description of supercontinuum generation,” Opt. Lett. 33, 2833–2835 (2008).
[CrossRef]

2007 (4)

S. Babin, D. Churkin, A. Ismagulov, S. Kablukov, and E. Podivilov, “Four-wave-mixing-induced turbulent spectral broadening in a long Raman fiber laser,” J. Opt. Soc. Am. B 24, 1729–1738 (2007).
[CrossRef]

S. Lagrange, H. R. Jauslin, and A. Picozzi, “Thermalization of the dispersive three-wave interaction,” Europhys. Lett. 79, 64001 (2007).
[CrossRef]

S. Skupin, M. Saffman, and W. Krolikowski, “Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor,” Phys. Rev. Lett. 98, 263902 (2007).
[CrossRef]

N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, “Shocks in nonlocal media,” Phys. Rev. Lett. 99, 043903 (2007).
[CrossRef]

2006 (5)

M. Wu, P. Krivosik, B. A. Kalinikos, and C. E. Patton, “Random generation of coherent solitary waves from incoherent waves,” Phys. Rev. Lett. 96, 227202 (2006).
[CrossRef]

O. Cohen, H. Buljan, T. Schwartz, J. Fleischer, and M. Segev, “Incoherent solitons in instantaneous nonlocal nonlinear media,” Phys. Rev. E 73, 015601 (2006).
[CrossRef]

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, “Stability of two-dimensional spatial solitons in nonlocal nonlinear media,” Phys. Rev. E 73, 066603 (2006).
[CrossRef]

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, “Observation of attraction between dark solitons,” Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef]

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity locking of incoherent nonlinear wave packets,” Phys. Rev. Lett. 97, 033902 (2006).
[CrossRef]

2005 (4)

A. Picozzi and P. Aschieri, “Influence of dispersion on the resonant interaction between three incoherent waves,” Phys. Rev. E 72, 046606 (2005).
[CrossRef]

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Observation of random-phase lattice solitons,” Nature (London) 433, 500–503(2005).
[CrossRef]

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

A. Sauter, S. Pitois, G. Millot, and A. Picozzi, “Incoherent modulation instability in instantaneous nonlinear Kerr media,” Opt. Lett. 30, 2143–2145 (2005).
[CrossRef]

2004 (8)

A. Picozzi, “Entropy and degree of polarization for nonlinear optical waves,” Opt. Lett. 29, 1653–1655 (2004).
[CrossRef]

A. Picozzi and M. Haelterman, “Condensation in Hamiltonian parametric wave interaction,” Phys. Rev. Lett. 92, 103901 (2004).
[CrossRef]

W. Krolikowski, O. Bang, and J. Wyller, “Nonlocal incoherent solitons,” Phys. Rev. E 70, 036617 (2004).
[CrossRef]

H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides, “Random-phase solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef]

V. E. Zakharov, F. Dias, and A. Pushkarev, “One-dimensional wave turbulence,” Phys. Rep. 398, 1–65 (2004).
[CrossRef]

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[CrossRef]

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, “Incoherent solitons in instantaneous response nonlinear media,” Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef]

C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef]

2003 (5)

C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef]

B. Rumpf and A. C. Newell, “Localization and coherence in nonintegrable systems,” Physica D 184, 162–191 (2003).
[CrossRef]

M. Onorato, A. Osborne, R. Fedele, and M. Serio, “Landau damping and coherent structures in narrow-banded 1+1 deep water gravity waves,” Phys. Rev. E 67, 046305 (2003).
[CrossRef]

M. Lisak, L. Helczynski, and D. Anderson, “Relation between different formalisms describing partially incoherent wave propagation in nonlinear optical media,” Opt. Commun. 220, 321–323 (2003).
[CrossRef]

J. Garnier, J.-P. Ayanides, and O. Morice, “Propagation of partially coherent light with the Maxwell–Debye equation,” J. Opt. Soc. Am. B 20, 1409–1417 (2003).
[CrossRef]

2002 (4)

A. Picozzi and M. Haelterman, “Hidden coherence along space-time trajectories in parametric wave mixing,” Phys. Rev. Lett. 88, 083901 (2002).
[CrossRef]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002).
[CrossRef]

J. Wyller, W. Krolikowski, O. Bang, and J. J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66, 066615 (2002).
[CrossRef]

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619 (2002).
[CrossRef]

2001 (7)

A. Picozzi and M. Haelterman, “Parametric three-wave soliton generated from incoherent light,” Phys. Rev. Lett. 86, 2010–2013 (2001).
[CrossRef]

B. Rumpf and A. C. Newell, “Coherent structures and entropy in constrained, modulationally unstable, nonintegrable systems,” Phys. Rev. Lett. 87, 054102 (2001).
[CrossRef]

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media,” Phys. Rev. E 63, 035601 (2001).
[CrossRef]

A. C. Newell, S. Nazarenko, and L. Biven, “Wave turbulence and intermittency,” Physica D 152, 520–550 (2001).
[CrossRef]

A. C. Newell and B. Rumpf, “Wave turbulence,” Ann. Rev. Fluids Mech. 43, 59–78 (2001).

M. J. Davis, S. A. Morgan, and K. Burnett, “Simulations of Bose fields at finite temperature,” Phys. Rev. Lett. 87, 160402 (2001).
[CrossRef]

M. Peccianti and G. Assanto, “Incoherent spatial solitary waves in nematic liquid crystals,” Opt. Lett. 26, 1791–1793 (2001).
[CrossRef]

2000 (3)

M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000).
[CrossRef]

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[CrossRef]

R. Jordan and C. Josserand, “Self-organization in nonlinear wave turbulence,” Phys. Rev. E 61, 1527–1539 (2000).
[CrossRef]

1999 (1)

O. Bang, D. Edmundson, and W. Krolikowski, “Collapse of incoherent light beams in inertial bulk Kerr media,” Phys. Rev. Lett. 83, 5479–5482 (1999).
[CrossRef]

1998 (3)

N. Akhmediev, W. Krolikowski, and A. W. Snyder, “Partially voherent solitons of variable shape,” Phys. Rev. Lett. 81, 4632–4635 (1998).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, Z. Chen, and M. Segev, “Theory of incoherent dark solitons,” Phys. Rev. Lett. 80, 5113–5116 (1998).
[CrossRef]

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998).
[CrossRef]

1997 (6)

M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media,” Phys. Rev. Lett. 78, 646–649 (1997).
[CrossRef]

M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature (London) 387, 880–883 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media” Phys. Rev. Lett. 78, 646–649 (1997).
[CrossRef]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
[CrossRef]

A. Snyder and D. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
[CrossRef]

1996 (1)

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[CrossRef]

1995 (1)

S. L. Musher, A. M. Rubenchik, and V. E. Zakharov, “Weak Langmuir turbulence,” Phys. Rep. 252, 177–274 (1995).
[CrossRef]

1992 (1)

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
[CrossRef]

1988 (1)

V. E. Zakharov, A. N. Pushkarev, V. F. Shvets, and V. V. Yan’kov, “Soliton turbulence,” JETP Lett. 48, 83–87 (1988).

1985 (1)

V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, “Hamiltonian approach to the description of non-linear plasma phenomena,” Phys. Rep. 129, 285–366 (1985).
[CrossRef]

1979 (2)

C. Montes, “Photon soliton and fine structure due to nonlinear Compton scattering,” Phys. Rev. A 20, 1081–1095 (1979).
[CrossRef]

C. Montes, J. Peyraud, and M. Hénon, “One-dimensional boson soliton collisions,” Phys. Fluids 22, 176–182 (1979).
[CrossRef]

1978 (1)

A. G. Litvak and A. M. Sergeev, “One dimensional collapse of plasma waves,” JETP Lett. 27, 517–520 (1978).

1977 (1)

A. Hasegawa, “Envelope soliton of random phase waves,” Phys. Fluids 20, 2155–2156 (1977).
[CrossRef]

1975 (2)

A. Hasegawa, “Dynamics of an ensemble of plane waves in nonlinear dispersive media,” Phys. Fluids 18, 77–79 (1975).
[CrossRef]

V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, “Weak Langmuir turbulence of an isothermal plasma,” Sov. Phys. JETP 42, 80–86 (1975).

1974 (1)

G. A. Pasmanik, “Self-interaction of incoherent light beams,” Sov. Phys. JETP 39, 234–238 (1974).

1972 (1)

Y. B. Zel’dovich, E. V. Levich, and R. A. Syunyaev, “Stimulated Compton interaction between Maxwellian electrons and spectrally narrow radiation,” Sov. Phys. JETP 35, 733–740 (1972).

1968 (1)

A. C. Newell, “The closure problem in a system of random gravity waves,” Rev. Geophys. 6, 1–31 (1968).
[CrossRef]

1963 (1)

K. Hasselmann, “On the non-linear energy transfer in a gravity-wave spectrum. Part 2. Conservation theorems; wave-particle analogy; irreversibility,” J. Fluid Mech. 15, 273–281 (1963).
[CrossRef]

1962 (1)

K. Hasselmann, “On the non-linear energy transfer in a gravity-wave spectrum. Part 1. General theory,” J. Fluid Mech. 12, 481–500 (1962).
[CrossRef]

1957 (1)

I. B. Bernstein, J. M. Green, and M. D. Kruskal, “Exact nonlinear plasma oscillations,” Phys. Rev. 108, 546–550 (1957).
[CrossRef]

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons : From Fibers to Photonic Crystals (Academic Press, 2003).

Akhmediev, N.

N. Akhmediev, W. Krolikowski, and A. W. Snyder, “Partially voherent solitons of variable shape,” Phys. Rev. Lett. 81, 4632–4635 (1998).
[CrossRef]

Anderson, D.

T. Hansson, M. Lisak, and D. Anderson, “Integrability and conservation laws for the nonlinear evolution equations of partially coherent waves in noninstantaneous Kerr media,” Phys. Rev. Lett. 108, 063901 (2012).
[CrossRef]

T. Hansson, D. Anderson, M. Lisak, V. E. Semenov, and U. Osterberg, “Propagation of partially coherent light beams with parabolic intensity distribution in noninstantaneous nonlinear Kerr media,” J. Opt. Soc. Am. B 25, 1780–1785 (2008).
[CrossRef]

M. Lisak, L. Helczynski, and D. Anderson, “Relation between different formalisms describing partially incoherent wave propagation in nonlinear optical media,” Opt. Commun. 220, 321–323 (2003).
[CrossRef]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Angelani, L.

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, “Condensation in disordered lasers: theory, 3D+1 simulations, and experiments,” Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

Ania-Castanon, J. D.

S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castanon, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nat. Photon. 4, 231–235 (2010).
[CrossRef]

S. A. Babin, V. Karalekas, E. V. Podivilov, V. K. Mezentsev, P. Harper, J. D. Ania-Castanon, and S. K. Turitsyn, “Turbulent broadening of optical spectra in ultralong Raman fiber lasers,” Phys. Rev. A 77, 033803 (2008).
[CrossRef]

Aschieri, P.

G. Strömqvist, V. Pasiskevicius, C. Canalias, P. Aschieri, A. Picozzi, and C. Montes, “Temporal coherence in mirrorless optical parametric oscillators,” J. Opt. Soc. Am. B 29, 1194–1202 (2012).
[CrossRef]

P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi, “Condensation and thermalization of classsical optical waves in a waveguide,” Phys. Rev. A 83, 033838 (2011).
[CrossRef]

A. Picozzi and P. Aschieri, “Influence of dispersion on the resonant interaction between three incoherent waves,” Phys. Rev. E 72, 046606 (2005).
[CrossRef]

Assanto, G.

C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef]

C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef]

M. Peccianti and G. Assanto, “Incoherent spatial solitary waves in nematic liquid crystals,” Opt. Lett. 26, 1791–1793 (2001).
[CrossRef]

Ayanides, J.-P.

Babin, S.

Babin, S. A.

S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castanon, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nat. Photon. 4, 231–235 (2010).
[CrossRef]

S. A. Babin, V. Karalekas, E. V. Podivilov, V. K. Mezentsev, P. Harper, J. D. Ania-Castanon, and S. K. Turitsyn, “Turbulent broadening of optical spectra in ultralong Raman fiber lasers,” Phys. Rev. A 77, 033803 (2008).
[CrossRef]

Bang, O.

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, “Observation of attraction between dark solitons,” Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef]

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, “Stability of two-dimensional spatial solitons in nonlocal nonlinear media,” Phys. Rev. E 73, 066603 (2006).
[CrossRef]

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[CrossRef]

W. Krolikowski, O. Bang, and J. Wyller, “Nonlocal incoherent solitons,” Phys. Rev. E 70, 036617 (2004).
[CrossRef]

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619 (2002).
[CrossRef]

J. Wyller, W. Krolikowski, O. Bang, and J. J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66, 066615 (2002).
[CrossRef]

O. Bang, D. Edmundson, and W. Krolikowski, “Collapse of incoherent light beams in inertial bulk Kerr media,” Phys. Rev. Lett. 83, 5479–5482 (1999).
[CrossRef]

Barsi, C.

C. Barsi, W. Wan, and J. W. Fleischer, “Imaging through nonlinear media via digital holography,” Nat. Photon. 3, 211–215 (2009).
[CrossRef]

Bartal, G.

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Observation of random-phase lattice solitons,” Nature (London) 433, 500–503(2005).
[CrossRef]

Barviau, B.

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in photonic crystal fiber,” Phys. Rev. E 84, 066605 (2011).
[CrossRef]

B. Barviau, B. Kibler, A. Kudlinski, A. Mussot, G. Millot, and A. Picozzi, “Experimental signature of optical wave thermalization through supercontinuum generation in photonic crystal fiber,” Opt. Express 17, 7392–7406 (2009).
[CrossRef]

B. Barviau, B. Kibler, and A. Picozzi, “Wave-turbulence approach of supercontinuum generation: influence of self-steepening and higher-order dispersion,” Phys. Rev. A 79, 063840 (2009).
[CrossRef]

B. Barviau, B. Kibler, S. Coen, and A. Picozzi, “Towards a thermodynamic description of supercontinuum generation,” Opt. Lett. 33, 2833–2835 (2008).
[CrossRef]

Bekker, A.

Bernstein, I. B.

I. B. Bernstein, J. M. Green, and M. D. Kruskal, “Exact nonlinear plasma oscillations,” Phys. Rev. 108, 546–550 (1957).
[CrossRef]

Biancalana, F.

C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, “Highly noninstantaneous solitons in liquid-core photonic crystal fibers,” Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

Biven, L.

A. C. Newell, S. Nazarenko, and L. Biven, “Wave turbulence and intermittency,” Physica D 152, 520–550 (2001).
[CrossRef]

Bortolozzo, U.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic Press, 2008).

Bromberg, Y.

Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury Brown and Twiss interferometry with interacting photons,” Nat. Photon. 4, 721–726 (2010).
[CrossRef]

Buljan, H.

O. Cohen, H. Buljan, T. Schwartz, J. Fleischer, and M. Segev, “Incoherent solitons in instantaneous nonlocal nonlinear media,” Phys. Rev. E 73, 015601 (2006).
[CrossRef]

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Observation of random-phase lattice solitons,” Nature (London) 433, 500–503(2005).
[CrossRef]

H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides, “Random-phase solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef]

Burnett, K.

M. J. Davis, S. A. Morgan, and K. Burnett, “Simulations of Bose fields at finite temperature,” Phys. Rev. Lett. 87, 160402 (2001).
[CrossRef]

Campa, A.

A. Campa, T. Dauxois, and S. Ruffo, “Statistical mechanics and dynamics of solvable models with long-range interactions,” Phys. Rep. 480, 57–159 (2009).
[CrossRef]

Canalias, C.

Carmon, T.

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Observation of random-phase lattice solitons,” Nature (London) 433, 500–503(2005).
[CrossRef]

Chen, Z.

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, Z. Chen, and M. Segev, “Theory of incoherent dark solitons,” Phys. Rev. Lett. 80, 5113–5116 (1998).
[CrossRef]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[CrossRef]

Christodoulides, D. N.

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Observation of random-phase lattice solitons,” Nature (London) 433, 500–503(2005).
[CrossRef]

H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides, “Random-phase solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef]

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media,” Phys. Rev. E 63, 035601 (2001).
[CrossRef]

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[CrossRef]

M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000).
[CrossRef]

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, Z. Chen, and M. Segev, “Theory of incoherent dark solitons,” Phys. Rev. Lett. 80, 5113–5116 (1998).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media” Phys. Rev. Lett. 78, 646–649 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media,” Phys. Rev. Lett. 78, 646–649 (1997).
[CrossRef]

M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
[CrossRef]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
[CrossRef]

Churkin, D.

Churkin, D. V.

S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castanon, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nat. Photon. 4, 231–235 (2010).
[CrossRef]

Coen, S.

Cohen, O.

C. Rotschild, T. Schwartz, O. Cohen, and M. Segev, “Incoherent spatial solitons in effectively-instantaneous nonlocal nonlinear media,” Nat. Photon. 2, 371–376 (2008).
[CrossRef]

O. Cohen, H. Buljan, T. Schwartz, J. Fleischer, and M. Segev, “Incoherent solitons in instantaneous nonlocal nonlinear media,” Phys. Rev. E 73, 015601 (2006).
[CrossRef]

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Observation of random-phase lattice solitons,” Nature (London) 433, 500–503(2005).
[CrossRef]

H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides, “Random-phase solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef]

Connaughton, C.

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

Conti, C.

C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, “Highly noninstantaneous solitons in liquid-core photonic crystal fibers,” Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef]

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, “Condensation in disordered lasers: theory, 3D+1 simulations, and experiments,” Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, “Shocks in nonlocal media,” Phys. Rev. Lett. 99, 043903 (2007).
[CrossRef]

C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef]

C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef]

Coskun, T.

M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000).
[CrossRef]

M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
[CrossRef]

Coskun, T. H.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media,” Phys. Rev. E 63, 035601 (2001).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, Z. Chen, and M. Segev, “Theory of incoherent dark solitons,” Phys. Rev. Lett. 80, 5113–5116 (1998).
[CrossRef]

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998).
[CrossRef]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media” Phys. Rev. Lett. 78, 646–649 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media,” Phys. Rev. Lett. 78, 646–649 (1997).
[CrossRef]

Crosignani, B.

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
[CrossRef]

Dauxois, T.

A. Campa, T. Dauxois, and S. Ruffo, “Statistical mechanics and dynamics of solvable models with long-range interactions,” Phys. Rep. 480, 57–159 (2009).
[CrossRef]

Davis, M. J.

M. J. Davis, S. A. Morgan, and K. Burnett, “Simulations of Bose fields at finite temperature,” Phys. Rev. Lett. 87, 160402 (2001).
[CrossRef]

Dias, F.

V. E. Zakharov, F. Dias, and A. Pushkarev, “One-dimensional wave turbulence,” Phys. Rep. 398, 1–65 (2004).
[CrossRef]

Doya, V.

P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi, “Condensation and thermalization of classsical optical waves in a waveguide,” Phys. Rev. A 83, 033838 (2011).
[CrossRef]

Dreischuh, A.

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, “Observation of attraction between dark solitons,” Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef]

Düring, G.

G. Düring, A. Picozzi, and S. Rica, “Breakdown of weak-turbulence and nonlinear wave condensation,” Physica D 238, 1524–1549 (2009).
[CrossRef]

Dylov, D. V.

D. V. Dylov and J. W. Fleischer, “Observation of all-optical bump-on-tail instability,” Phys. Rev. Lett. 100, 103903 (2008).
[CrossRef]

Edmundson, D.

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, “Stability of two-dimensional spatial solitons in nonlocal nonlinear media,” Phys. Rev. E 73, 066603 (2006).
[CrossRef]

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[CrossRef]

O. Bang, D. Edmundson, and W. Krolikowski, “Collapse of incoherent light beams in inertial bulk Kerr media,” Phys. Rev. Lett. 83, 5479–5482 (1999).
[CrossRef]

Efremidis, N. K.

H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides, “Random-phase solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef]

El-Taher, A. E.

S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castanon, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nat. Photon. 4, 231–235 (2010).
[CrossRef]

Eugenieva, E.

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[CrossRef]

Eugenieva, E. D.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media,” Phys. Rev. E 63, 035601 (2001).
[CrossRef]

Falkovich, G.

E. G. Turitsyna, G. Falkovich, V. K. Mezentsev, and S. K. Turitsyn, “Optical turbulence and spectral condensate in long-fiber lasers,” Phys. Rev. A 80, 031804 (2009).
[CrossRef]

V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I (Springer, 1992).

Fedele, R.

M. Onorato, A. Osborne, R. Fedele, and M. Serio, “Landau damping and coherent structures in narrow-banded 1+1 deep water gravity waves,” Phys. Rev. E 67, 046305 (2003).
[CrossRef]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Finot, C.

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374, 3585–3589 (2010).
[CrossRef]

Fischer, B.

R. Weill, B. Fischer, and O. Gat, “Light-mode condensation in actively-mode-locked lasers,” Phys. Rev. Lett. 104, 173901 (2010).
[CrossRef]

R. Weill, B. Levit, A. Bekker, O. Gat, and B. Fischer, “Laser light condensate: experimental demonstration of light-mode condensation in actively mode locked laser,” Opt. Express 18, 16520–16525 (2010).
[CrossRef]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
[CrossRef]

Fleischer, J.

O. Cohen, H. Buljan, T. Schwartz, J. Fleischer, and M. Segev, “Incoherent solitons in instantaneous nonlocal nonlinear media,” Phys. Rev. E 73, 015601 (2006).
[CrossRef]

Fleischer, J. W.

C. Barsi, W. Wan, and J. W. Fleischer, “Imaging through nonlinear media via digital holography,” Nat. Photon. 3, 211–215 (2009).
[CrossRef]

D. V. Dylov and J. W. Fleischer, “Observation of all-optical bump-on-tail instability,” Phys. Rev. Lett. 100, 103903 (2008).
[CrossRef]

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Observation of random-phase lattice solitons,” Nature (London) 433, 500–503(2005).
[CrossRef]

H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides, “Random-phase solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef]

Fratalocchi, A.

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef]

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, “Condensation in disordered lasers: theory, 3D+1 simulations, and experiments,” Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

Garnier, J.

B. Kibler, C. Michel, J. Garnier, and A. Picozzi, “Temporal dynamics of incoherent waves in noninstantaneous response nonlinear Kerr media,” Opt. Lett. 37, 2472–2474 (2012).

P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi, “Condensation and thermalization of classsical optical waves in a waveguide,” Phys. Rev. A 83, 033838 (2011).
[CrossRef]

A. Picozzi and J. Garnier, “Incoherent soliton turbulence in nonlocal nonlinear media,” Phys. Rev. Lett. 107, 233901(2011).
[CrossRef]

J. Garnier and A. Picozzi, “Unified kinetic formulation of incoherent waves propagating in nonlinear media with noninstantaneous response,” Phys. Rev. A 81, 033831 (2010).
[CrossRef]

J. Garnier, J.-P. Ayanides, and O. Morice, “Propagation of partially coherent light with the Maxwell–Debye equation,” J. Opt. Soc. Am. B 20, 1409–1417 (2003).
[CrossRef]

Gat, O.

Ghofraniha, N.

N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, “Shocks in nonlocal media,” Phys. Rev. Lett. 99, 043903 (2007).
[CrossRef]

Green, J. M.

I. B. Bernstein, J. M. Green, and M. D. Kruskal, “Exact nonlinear plasma oscillations,” Phys. Rev. 108, 546–550 (1957).
[CrossRef]

Haelterman, M.

C. Michel, M. Haelterman, P. Suret, S. Randoux, R. Kaiser, and A. Picozzi, “Thermalization and condensation in an incoherently pumped passive optical cavity,” Phys. Rev. A 84, 033848 (2011).
[CrossRef]

A. Picozzi and M. Haelterman, “Condensation in Hamiltonian parametric wave interaction,” Phys. Rev. Lett. 92, 103901 (2004).
[CrossRef]

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, “Incoherent solitons in instantaneous response nonlinear media,” Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef]

A. Picozzi and M. Haelterman, “Hidden coherence along space-time trajectories in parametric wave mixing,” Phys. Rev. Lett. 88, 083901 (2002).
[CrossRef]

A. Picozzi and M. Haelterman, “Parametric three-wave soliton generated from incoherent light,” Phys. Rev. Lett. 86, 2010–2013 (2001).
[CrossRef]

Hall, B.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Hammani, K.

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374, 3585–3589 (2010).
[CrossRef]

Hansson, T.

T. Hansson, M. Lisak, and D. Anderson, “Integrability and conservation laws for the nonlinear evolution equations of partially coherent waves in noninstantaneous Kerr media,” Phys. Rev. Lett. 108, 063901 (2012).
[CrossRef]

T. Hansson, D. Anderson, M. Lisak, V. E. Semenov, and U. Osterberg, “Propagation of partially coherent light beams with parabolic intensity distribution in noninstantaneous nonlinear Kerr media,” J. Opt. Soc. Am. B 25, 1780–1785 (2008).
[CrossRef]

Harper, P.

S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castanon, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nat. Photon. 4, 231–235 (2010).
[CrossRef]

S. A. Babin, V. Karalekas, E. V. Podivilov, V. K. Mezentsev, P. Harper, J. D. Ania-Castanon, and S. K. Turitsyn, “Turbulent broadening of optical spectra in ultralong Raman fiber lasers,” Phys. Rev. A 77, 033803 (2008).
[CrossRef]

Hasegawa, A.

A. Hasegawa, “Envelope soliton of random phase waves,” Phys. Fluids 20, 2155–2156 (1977).
[CrossRef]

A. Hasegawa, “Dynamics of an ensemble of plane waves in nonlinear dispersive media,” Phys. Fluids 18, 77–79 (1975).
[CrossRef]

Hasselmann, K.

K. Hasselmann, “On the non-linear energy transfer in a gravity-wave spectrum. Part 2. Conservation theorems; wave-particle analogy; irreversibility,” J. Fluid Mech. 15, 273–281 (1963).
[CrossRef]

K. Hasselmann, “On the non-linear energy transfer in a gravity-wave spectrum. Part 1. General theory,” J. Fluid Mech. 12, 481–500 (1962).
[CrossRef]

Helczynski, L.

M. Lisak, L. Helczynski, and D. Anderson, “Relation between different formalisms describing partially incoherent wave propagation in nonlinear optical media,” Opt. Commun. 220, 321–323 (2003).
[CrossRef]

Hénon, M.

C. Montes, J. Peyraud, and M. Hénon, “One-dimensional boson soliton collisions,” Phys. Fluids 22, 176–182 (1979).
[CrossRef]

Hsu, W. L.

Ismagulov, A.

Jauslin, H. R.

P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, “Anomalous thermalization of nonlinear wave systems,” Phys. Rev. Lett. 104, 054101 (2010).
[CrossRef]

C. Michel, P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, “Influence of third-order dispersion on the propagation of incoherent light in optical fibers,” Opt. Lett. 35, 2367–2369 (2010).
[CrossRef]

S. Lagrange, H. R. Jauslin, and A. Picozzi, “Thermalization of the dispersive three-wave interaction,” Europhys. Lett. 79, 64001 (2007).
[CrossRef]

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity locking of incoherent nonlinear wave packets,” Phys. Rev. Lett. 97, 033902 (2006).
[CrossRef]

Jordan, R.

R. Jordan and C. Josserand, “Self-organization in nonlinear wave turbulence,” Phys. Rev. E 61, 1527–1539 (2000).
[CrossRef]

Josserand, C.

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

R. Jordan and C. Josserand, “Self-organization in nonlinear wave turbulence,” Phys. Rev. E 61, 1527–1539 (2000).
[CrossRef]

Kablukov, S.

Kablukov, S. I.

S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castanon, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nat. Photon. 4, 231–235 (2010).
[CrossRef]

Kaiser, R.

C. Michel, M. Haelterman, P. Suret, S. Randoux, R. Kaiser, and A. Picozzi, “Thermalization and condensation in an incoherently pumped passive optical cavity,” Phys. Rev. A 84, 033848 (2011).
[CrossRef]

Kalinikos, B. A.

M. Wu, P. Krivosik, B. A. Kalinikos, and C. E. Patton, “Random generation of coherent solitary waves from incoherent waves,” Phys. Rev. Lett. 96, 227202 (2006).
[CrossRef]

Karalekas, V.

S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castanon, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nat. Photon. 4, 231–235 (2010).
[CrossRef]

S. A. Babin, V. Karalekas, E. V. Podivilov, V. K. Mezentsev, P. Harper, J. D. Ania-Castanon, and S. K. Turitsyn, “Turbulent broadening of optical spectra in ultralong Raman fiber lasers,” Phys. Rev. A 77, 033803 (2008).
[CrossRef]

Kibler, B.

B. Kibler, C. Michel, J. Garnier, and A. Picozzi, “Temporal dynamics of incoherent waves in noninstantaneous response nonlinear Kerr media,” Opt. Lett. 37, 2472–2474 (2012).

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in photonic crystal fiber,” Phys. Rev. E 84, 066605 (2011).
[CrossRef]

C. Michel, B. Kibler, and A. Picozzi, “Discrete spectral incoherent solitons in nonlinear media with noninstantaneous response,” Phys. Rev. A 83, 023806 (2011).
[CrossRef]

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374, 3585–3589 (2010).
[CrossRef]

B. Barviau, B. Kibler, and A. Picozzi, “Wave-turbulence approach of supercontinuum generation: influence of self-steepening and higher-order dispersion,” Phys. Rev. A 79, 063840 (2009).
[CrossRef]

B. Barviau, B. Kibler, A. Kudlinski, A. Mussot, G. Millot, and A. Picozzi, “Experimental signature of optical wave thermalization through supercontinuum generation in photonic crystal fiber,” Opt. Express 17, 7392–7406 (2009).
[CrossRef]

B. Barviau, B. Kibler, S. Coen, and A. Picozzi, “Towards a thermodynamic description of supercontinuum generation,” Opt. Lett. 33, 2833–2835 (2008).
[CrossRef]

Kip, D.

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[CrossRef]

Kivshar, Y. S.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons : From Fibers to Photonic Crystals (Academic Press, 2003).

Klaers, J.

J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468, 545–548 (2010).
[CrossRef]

Koplow, J. P.

Krivosik, P.

M. Wu, P. Krivosik, B. A. Kalinikos, and C. E. Patton, “Random generation of coherent solitary waves from incoherent waves,” Phys. Rev. Lett. 96, 227202 (2006).
[CrossRef]

Krolikowski, W.

S. Skupin, M. Saffman, and W. Krolikowski, “Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor,” Phys. Rev. Lett. 98, 263902 (2007).
[CrossRef]

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, “Stability of two-dimensional spatial solitons in nonlocal nonlinear media,” Phys. Rev. E 73, 066603 (2006).
[CrossRef]

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, “Observation of attraction between dark solitons,” Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef]

W. Krolikowski, O. Bang, and J. Wyller, “Nonlocal incoherent solitons,” Phys. Rev. E 70, 036617 (2004).
[CrossRef]

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[CrossRef]

J. Wyller, W. Krolikowski, O. Bang, and J. J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66, 066615 (2002).
[CrossRef]

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619 (2002).
[CrossRef]

O. Bang, D. Edmundson, and W. Krolikowski, “Collapse of incoherent light beams in inertial bulk Kerr media,” Phys. Rev. Lett. 83, 5479–5482 (1999).
[CrossRef]

N. Akhmediev, W. Krolikowski, and A. W. Snyder, “Partially voherent solitons of variable shape,” Phys. Rev. Lett. 81, 4632–4635 (1998).
[CrossRef]

Kruskal, M. D.

I. B. Bernstein, J. M. Green, and M. D. Kruskal, “Exact nonlinear plasma oscillations,” Phys. Rev. 108, 546–550 (1957).
[CrossRef]

Kudlinski, A.

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in photonic crystal fiber,” Phys. Rev. E 84, 066605 (2011).
[CrossRef]

B. Barviau, B. Kibler, A. Kudlinski, A. Mussot, G. Millot, and A. Picozzi, “Experimental signature of optical wave thermalization through supercontinuum generation in photonic crystal fiber,” Opt. Express 17, 7392–7406 (2009).
[CrossRef]

L’vov, V. S.

V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I (Springer, 1992).

Lagrange, S.

S. Lagrange, H. R. Jauslin, and A. Picozzi, “Thermalization of the dispersive three-wave interaction,” Europhys. Lett. 79, 64001 (2007).
[CrossRef]

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity locking of incoherent nonlinear wave packets,” Phys. Rev. Lett. 97, 033902 (2006).
[CrossRef]

Lahaye, T.

T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, “The physics of dipolar bosonic quantum gases,” Rep. Prog. Phys. 72, 126401 (2009).
[CrossRef]

Lahini, Y.

Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury Brown and Twiss interferometry with interacting photons,” Nat. Photon. 4, 721–726 (2010).
[CrossRef]

Y. Silberberg, Y. Lahini, E. Small, and R. Morandotti, “Universal correlations in a nonlinear periodic 1D system,” Phys. Rev. Lett. 102, 233904 (2009).
[CrossRef]

Laurie, J.

Leonetti, M.

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, “Condensation in disordered lasers: theory, 3D+1 simulations, and experiments,” Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

Levich, E. V.

Y. B. Zel’dovich, E. V. Levich, and R. A. Syunyaev, “Stimulated Compton interaction between Maxwellian electrons and spectrally narrow radiation,” Sov. Phys. JETP 35, 733–740 (1972).

Levit, B.

Lewenstein, M.

T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, “The physics of dipolar bosonic quantum gases,” Rep. Prog. Phys. 72, 126401 (2009).
[CrossRef]

Lisak, M.

T. Hansson, M. Lisak, and D. Anderson, “Integrability and conservation laws for the nonlinear evolution equations of partially coherent waves in noninstantaneous Kerr media,” Phys. Rev. Lett. 108, 063901 (2012).
[CrossRef]

T. Hansson, D. Anderson, M. Lisak, V. E. Semenov, and U. Osterberg, “Propagation of partially coherent light beams with parabolic intensity distribution in noninstantaneous nonlinear Kerr media,” J. Opt. Soc. Am. B 25, 1780–1785 (2008).
[CrossRef]

M. Lisak, L. Helczynski, and D. Anderson, “Relation between different formalisms describing partially incoherent wave propagation in nonlinear optical media,” Opt. Commun. 220, 321–323 (2003).
[CrossRef]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Litvak, A. G.

A. G. Litvak and A. M. Sergeev, “One dimensional collapse of plasma waves,” JETP Lett. 27, 517–520 (1978).

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Menotti, C.

T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, “The physics of dipolar bosonic quantum gases,” Rep. Prog. Phys. 72, 126401 (2009).
[CrossRef]

Mezentsev, V. K.

E. G. Turitsyna, G. Falkovich, V. K. Mezentsev, and S. K. Turitsyn, “Optical turbulence and spectral condensate in long-fiber lasers,” Phys. Rev. A 80, 031804 (2009).
[CrossRef]

S. A. Babin, V. Karalekas, E. V. Podivilov, V. K. Mezentsev, P. Harper, J. D. Ania-Castanon, and S. K. Turitsyn, “Turbulent broadening of optical spectra in ultralong Raman fiber lasers,” Phys. Rev. A 77, 033803 (2008).
[CrossRef]

Michel, C.

B. Kibler, C. Michel, J. Garnier, and A. Picozzi, “Temporal dynamics of incoherent waves in noninstantaneous response nonlinear Kerr media,” Opt. Lett. 37, 2472–2474 (2012).

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in photonic crystal fiber,” Phys. Rev. E 84, 066605 (2011).
[CrossRef]

P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi, “Condensation and thermalization of classsical optical waves in a waveguide,” Phys. Rev. A 83, 033838 (2011).
[CrossRef]

C. Michel, M. Haelterman, P. Suret, S. Randoux, R. Kaiser, and A. Picozzi, “Thermalization and condensation in an incoherently pumped passive optical cavity,” Phys. Rev. A 84, 033848 (2011).
[CrossRef]

C. Michel, B. Kibler, and A. Picozzi, “Discrete spectral incoherent solitons in nonlinear media with noninstantaneous response,” Phys. Rev. A 83, 023806 (2011).
[CrossRef]

C. Michel, P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, “Influence of third-order dispersion on the propagation of incoherent light in optical fibers,” Opt. Lett. 35, 2367–2369 (2010).
[CrossRef]

Millot, G.

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in photonic crystal fiber,” Phys. Rev. E 84, 066605 (2011).
[CrossRef]

B. Barviau, B. Kibler, A. Kudlinski, A. Mussot, G. Millot, and A. Picozzi, “Experimental signature of optical wave thermalization through supercontinuum generation in photonic crystal fiber,” Opt. Express 17, 7392–7406 (2009).
[CrossRef]

A. Picozzi, S. Pitois, and G. Millot, “Spectral incoherent solitons: a localized soliton behavior in the frequency domain,” Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef]

A. Sauter, S. Pitois, G. Millot, and A. Picozzi, “Incoherent modulation instability in instantaneous nonlinear Kerr media,” Opt. Lett. 30, 2143–2145 (2005).
[CrossRef]

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, “Incoherent solitons in instantaneous response nonlinear media,” Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef]

Mitchell, D.

A. Snyder and D. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
[CrossRef]

Mitchell, M.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media,” Phys. Rev. E 63, 035601 (2001).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, Z. Chen, and M. Segev, “Theory of incoherent dark solitons,” Phys. Rev. Lett. 80, 5113–5116 (1998).
[CrossRef]

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998).
[CrossRef]

M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
[CrossRef]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
[CrossRef]

M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature (London) 387, 880–883 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media” Phys. Rev. Lett. 78, 646–649 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media,” Phys. Rev. Lett. 78, 646–649 (1997).
[CrossRef]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[CrossRef]

Montes, C.

G. Strömqvist, V. Pasiskevicius, C. Canalias, P. Aschieri, A. Picozzi, and C. Montes, “Temporal coherence in mirrorless optical parametric oscillators,” J. Opt. Soc. Am. B 29, 1194–1202 (2012).
[CrossRef]

C. Montes, “Photon soliton and fine structure due to nonlinear Compton scattering,” Phys. Rev. A 20, 1081–1095 (1979).
[CrossRef]

C. Montes, J. Peyraud, and M. Hénon, “One-dimensional boson soliton collisions,” Phys. Fluids 22, 176–182 (1979).
[CrossRef]

Moore, S. W.

Morandotti, R.

Y. Silberberg, Y. Lahini, E. Small, and R. Morandotti, “Universal correlations in a nonlinear periodic 1D system,” Phys. Rev. Lett. 102, 233904 (2009).
[CrossRef]

Morgan, S. A.

M. J. Davis, S. A. Morgan, and K. Burnett, “Simulations of Bose fields at finite temperature,” Phys. Rev. Lett. 87, 160402 (2001).
[CrossRef]

Morice, O.

Musher, S. L.

S. L. Musher, A. M. Rubenchik, and V. E. Zakharov, “Weak Langmuir turbulence,” Phys. Rep. 252, 177–274 (1995).
[CrossRef]

V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, “Hamiltonian approach to the description of non-linear plasma phenomena,” Phys. Rep. 129, 285–366 (1985).
[CrossRef]

V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, “Weak Langmuir turbulence of an isothermal plasma,” Sov. Phys. JETP 42, 80–86 (1975).

Musslimani, Z. H.

H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides, “Random-phase solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef]

Mussot, A.

Nazarenko, S.

U. Bortolozzo, J. Laurie, S. Nazarenko, and S. Residori, “Optical wave turbulence and the condensation of light,” J. Opt. Soc. Am. B 26, 2280–2284 (2009).
[CrossRef]

A. C. Newell, S. Nazarenko, and L. Biven, “Wave turbulence and intermittency,” Physica D 152, 520–550 (2001).
[CrossRef]

S. Nazarenko, Wave Turbulence, Lectures Notes in Physics825 (Springer, 2011).

Neshev, D.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[CrossRef]

Neshev, D. N.

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, “Observation of attraction between dark solitons,” Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef]

Newell, A. C.

B. Rumpf and A. C. Newell, “Localization and coherence in nonintegrable systems,” Physica D 184, 162–191 (2003).
[CrossRef]

A. C. Newell and B. Rumpf, “Wave turbulence,” Ann. Rev. Fluids Mech. 43, 59–78 (2001).

B. Rumpf and A. C. Newell, “Coherent structures and entropy in constrained, modulationally unstable, nonintegrable systems,” Phys. Rev. Lett. 87, 054102 (2001).
[CrossRef]

A. C. Newell, S. Nazarenko, and L. Biven, “Wave turbulence and intermittency,” Physica D 152, 520–550 (2001).
[CrossRef]

A. C. Newell, “The closure problem in a system of random gravity waves,” Rev. Geophys. 6, 1–31 (1968).
[CrossRef]

Nikolov, N. I.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[CrossRef]

Onorato, M.

M. Onorato, A. Osborne, R. Fedele, and M. Serio, “Landau damping and coherent structures in narrow-banded 1+1 deep water gravity waves,” Phys. Rev. E 67, 046305 (2003).
[CrossRef]

Osborne, A.

M. Onorato, A. Osborne, R. Fedele, and M. Serio, “Landau damping and coherent structures in narrow-banded 1+1 deep water gravity waves,” Phys. Rev. E 67, 046305 (2003).
[CrossRef]

Osterberg, U.

Pasiskevicius, V.

Pasmanik, G. A.

G. A. Pasmanik, “Self-interaction of incoherent light beams,” Sov. Phys. JETP 39, 234–238 (1974).

Patton, C. E.

M. Wu, P. Krivosik, B. A. Kalinikos, and C. E. Patton, “Random generation of coherent solitary waves from incoherent waves,” Phys. Rev. Lett. 96, 227202 (2006).
[CrossRef]

Peccianti, M.

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef]

C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef]

C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef]

M. Peccianti and G. Assanto, “Incoherent spatial solitary waves in nematic liquid crystals,” Opt. Lett. 26, 1791–1793 (2001).
[CrossRef]

Petersen, D. E.

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, “Observation of attraction between dark solitons,” Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef]

Peyraud, J.

C. Montes, J. Peyraud, and M. Hénon, “One-dimensional boson soliton collisions,” Phys. Fluids 22, 176–182 (1979).
[CrossRef]

Pfau, T.

T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, “The physics of dipolar bosonic quantum gases,” Rep. Prog. Phys. 72, 126401 (2009).
[CrossRef]

Picozzi, A.

B. Kibler, C. Michel, J. Garnier, and A. Picozzi, “Temporal dynamics of incoherent waves in noninstantaneous response nonlinear Kerr media,” Opt. Lett. 37, 2472–2474 (2012).

G. Strömqvist, V. Pasiskevicius, C. Canalias, P. Aschieri, A. Picozzi, and C. Montes, “Temporal coherence in mirrorless optical parametric oscillators,” J. Opt. Soc. Am. B 29, 1194–1202 (2012).
[CrossRef]

P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi, “Condensation and thermalization of classsical optical waves in a waveguide,” Phys. Rev. A 83, 033838 (2011).
[CrossRef]

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in photonic crystal fiber,” Phys. Rev. E 84, 066605 (2011).
[CrossRef]

P. Suret, A. Picozzi, and S. Randoux, “Wave turbulence in integrable systems: nonlinear propagation of incoherent optical waves in single-mode fibers,” Opt. Express 19, 17852–17863 (2011).
[CrossRef]

A. Picozzi and J. Garnier, “Incoherent soliton turbulence in nonlocal nonlinear media,” Phys. Rev. Lett. 107, 233901(2011).
[CrossRef]

C. Michel, M. Haelterman, P. Suret, S. Randoux, R. Kaiser, and A. Picozzi, “Thermalization and condensation in an incoherently pumped passive optical cavity,” Phys. Rev. A 84, 033848 (2011).
[CrossRef]

C. Michel, B. Kibler, and A. Picozzi, “Discrete spectral incoherent solitons in nonlinear media with noninstantaneous response,” Phys. Rev. A 83, 023806 (2011).
[CrossRef]

J. Garnier and A. Picozzi, “Unified kinetic formulation of incoherent waves propagating in nonlinear media with noninstantaneous response,” Phys. Rev. A 81, 033831 (2010).
[CrossRef]

P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, “Anomalous thermalization of nonlinear wave systems,” Phys. Rev. Lett. 104, 054101 (2010).
[CrossRef]

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374, 3585–3589 (2010).
[CrossRef]

C. Michel, P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, “Influence of third-order dispersion on the propagation of incoherent light in optical fibers,” Opt. Lett. 35, 2367–2369 (2010).
[CrossRef]

B. Barviau, B. Kibler, A. Kudlinski, A. Mussot, G. Millot, and A. Picozzi, “Experimental signature of optical wave thermalization through supercontinuum generation in photonic crystal fiber,” Opt. Express 17, 7392–7406 (2009).
[CrossRef]

B. Barviau, B. Kibler, and A. Picozzi, “Wave-turbulence approach of supercontinuum generation: influence of self-steepening and higher-order dispersion,” Phys. Rev. A 79, 063840 (2009).
[CrossRef]

G. Düring, A. Picozzi, and S. Rica, “Breakdown of weak-turbulence and nonlinear wave condensation,” Physica D 238, 1524–1549 (2009).
[CrossRef]

A. Picozzi, “Spontaneous polarization induced by natural thermalization of incoherent light,” Opt. Express 16, 17171–17185 (2008).
[CrossRef]

A. Picozzi, S. Pitois, and G. Millot, “Spectral incoherent solitons: a localized soliton behavior in the frequency domain,” Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef]

B. Barviau, B. Kibler, S. Coen, and A. Picozzi, “Towards a thermodynamic description of supercontinuum generation,” Opt. Lett. 33, 2833–2835 (2008).
[CrossRef]

A. Picozzi and S. Rica, “Coherence absorption and condensation induced by thermalization of incoherent nonlinear fields,” Europhys. Lett. 84, 34004 (2008).
[CrossRef]

S. Lagrange, H. R. Jauslin, and A. Picozzi, “Thermalization of the dispersive three-wave interaction,” Europhys. Lett. 79, 64001 (2007).
[CrossRef]

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity locking of incoherent nonlinear wave packets,” Phys. Rev. Lett. 97, 033902 (2006).
[CrossRef]

A. Picozzi and P. Aschieri, “Influence of dispersion on the resonant interaction between three incoherent waves,” Phys. Rev. E 72, 046606 (2005).
[CrossRef]

A. Sauter, S. Pitois, G. Millot, and A. Picozzi, “Incoherent modulation instability in instantaneous nonlinear Kerr media,” Opt. Lett. 30, 2143–2145 (2005).
[CrossRef]

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

A. Picozzi and M. Haelterman, “Condensation in Hamiltonian parametric wave interaction,” Phys. Rev. Lett. 92, 103901 (2004).
[CrossRef]

A. Picozzi, “Entropy and degree of polarization for nonlinear optical waves,” Opt. Lett. 29, 1653–1655 (2004).
[CrossRef]

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, “Incoherent solitons in instantaneous response nonlinear media,” Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef]

A. Picozzi and M. Haelterman, “Hidden coherence along space-time trajectories in parametric wave mixing,” Phys. Rev. Lett. 88, 083901 (2002).
[CrossRef]

A. Picozzi and M. Haelterman, “Parametric three-wave soliton generated from incoherent light,” Phys. Rev. Lett. 86, 2010–2013 (2001).
[CrossRef]

Piskarskas, A.

A. Stabinis, V. Pyragaite, G. Tamoauskas, and A. Piskarskas, “Spectrum of second-harmonic radiation generated from incoherent light,” Phys. Rev. A 84, 043813 (2011).
[CrossRef]

A. Piskarskas, V. Pyragaite, and A. Stabinis, “Generation of coherent waves by frequency up-conversion and down-conversion of incoherent light,” Phys. Rev. A 82, 053817 (2010).
[CrossRef]

Pitois, S.

A. Picozzi, S. Pitois, and G. Millot, “Spectral incoherent solitons: a localized soliton behavior in the frequency domain,” Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef]

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity locking of incoherent nonlinear wave packets,” Phys. Rev. Lett. 97, 033902 (2006).
[CrossRef]

A. Sauter, S. Pitois, G. Millot, and A. Picozzi, “Incoherent modulation instability in instantaneous nonlinear Kerr media,” Opt. Lett. 30, 2143–2145 (2005).
[CrossRef]

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, “Incoherent solitons in instantaneous response nonlinear media,” Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef]

Podivilov, E.

Podivilov, E. V.

S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castanon, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nat. Photon. 4, 231–235 (2010).
[CrossRef]

S. A. Babin, V. Karalekas, E. V. Podivilov, V. K. Mezentsev, P. Harper, J. D. Ania-Castanon, and S. K. Turitsyn, “Turbulent broadening of optical spectra in ultralong Raman fiber lasers,” Phys. Rev. A 77, 033803 (2008).
[CrossRef]

Pomeau, Y.

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

Pushkarev, A.

V. E. Zakharov, F. Dias, and A. Pushkarev, “One-dimensional wave turbulence,” Phys. Rep. 398, 1–65 (2004).
[CrossRef]

Pushkarev, A. N.

V. E. Zakharov, A. N. Pushkarev, V. F. Shvets, and V. V. Yan’kov, “Soliton turbulence,” JETP Lett. 48, 83–87 (1988).

Pyragaite, V.

A. Stabinis, V. Pyragaite, G. Tamoauskas, and A. Piskarskas, “Spectrum of second-harmonic radiation generated from incoherent light,” Phys. Rev. A 84, 043813 (2011).
[CrossRef]

A. Piskarskas, V. Pyragaite, and A. Stabinis, “Generation of coherent waves by frequency up-conversion and down-conversion of incoherent light,” Phys. Rev. A 82, 053817 (2010).
[CrossRef]

Randoux, S.

C. Michel, M. Haelterman, P. Suret, S. Randoux, R. Kaiser, and A. Picozzi, “Thermalization and condensation in an incoherently pumped passive optical cavity,” Phys. Rev. A 84, 033848 (2011).
[CrossRef]

P. Suret, A. Picozzi, and S. Randoux, “Wave turbulence in integrable systems: nonlinear propagation of incoherent optical waves in single-mode fibers,” Opt. Express 19, 17852–17863 (2011).
[CrossRef]

C. Michel, P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, “Influence of third-order dispersion on the propagation of incoherent light in optical fibers,” Opt. Lett. 35, 2367–2369 (2010).
[CrossRef]

P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, “Anomalous thermalization of nonlinear wave systems,” Phys. Rev. Lett. 104, 054101 (2010).
[CrossRef]

Rasmussen, J. J.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[CrossRef]

J. Wyller, W. Krolikowski, O. Bang, and J. J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66, 066615 (2002).
[CrossRef]

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619 (2002).
[CrossRef]

Residori, S.

Rica, S.

G. Düring, A. Picozzi, and S. Rica, “Breakdown of weak-turbulence and nonlinear wave condensation,” Physica D 238, 1524–1549 (2009).
[CrossRef]

A. Picozzi and S. Rica, “Coherence absorption and condensation induced by thermalization of incoherent nonlinear fields,” Europhys. Lett. 84, 34004 (2008).
[CrossRef]

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

Rotschild, C.

C. Rotschild, T. Schwartz, O. Cohen, and M. Segev, “Incoherent spatial solitons in effectively-instantaneous nonlocal nonlinear media,” Nat. Photon. 2, 371–376 (2008).
[CrossRef]

Rubenchik, A. M.

S. L. Musher, A. M. Rubenchik, and V. E. Zakharov, “Weak Langmuir turbulence,” Phys. Rep. 252, 177–274 (1995).
[CrossRef]

V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, “Hamiltonian approach to the description of non-linear plasma phenomena,” Phys. Rep. 129, 285–366 (1985).
[CrossRef]

V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, “Weak Langmuir turbulence of an isothermal plasma,” Sov. Phys. JETP 42, 80–86 (1975).

Ruffo, S.

A. Campa, T. Dauxois, and S. Ruffo, “Statistical mechanics and dynamics of solvable models with long-range interactions,” Phys. Rep. 480, 57–159 (2009).
[CrossRef]

Rumpf, B.

B. Rumpf and A. C. Newell, “Localization and coherence in nonintegrable systems,” Physica D 184, 162–191 (2003).
[CrossRef]

A. C. Newell and B. Rumpf, “Wave turbulence,” Ann. Rev. Fluids Mech. 43, 59–78 (2001).

B. Rumpf and A. C. Newell, “Coherent structures and entropy in constrained, modulationally unstable, nonintegrable systems,” Phys. Rev. Lett. 87, 054102 (2001).
[CrossRef]

Ruocco, G.

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef]

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, “Condensation in disordered lasers: theory, 3D+1 simulations, and experiments,” Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, “Shocks in nonlocal media,” Phys. Rev. Lett. 99, 043903 (2007).
[CrossRef]

Russell, P. St. J.

C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, “Highly noninstantaneous solitons in liquid-core photonic crystal fibers,” Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

Saffman, M.

S. Skupin, M. Saffman, and W. Krolikowski, “Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor,” Phys. Rev. Lett. 98, 263902 (2007).
[CrossRef]

Santos, L.

T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, “The physics of dipolar bosonic quantum gases,” Rep. Prog. Phys. 72, 126401 (2009).
[CrossRef]

Sauter, A.

Schmidt, M. A.

C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, “Highly noninstantaneous solitons in liquid-core photonic crystal fibers,” Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

Schmitt, J.

J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468, 545–548 (2010).
[CrossRef]

Schroder, K. L.

Schwartz, T.

C. Rotschild, T. Schwartz, O. Cohen, and M. Segev, “Incoherent spatial solitons in effectively-instantaneous nonlocal nonlinear media,” Nat. Photon. 2, 371–376 (2008).
[CrossRef]

O. Cohen, H. Buljan, T. Schwartz, J. Fleischer, and M. Segev, “Incoherent solitons in instantaneous nonlocal nonlinear media,” Phys. Rev. E 73, 015601 (2006).
[CrossRef]

H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides, “Random-phase solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef]

Segev, M.

C. Rotschild, T. Schwartz, O. Cohen, and M. Segev, “Incoherent spatial solitons in effectively-instantaneous nonlocal nonlinear media,” Nat. Photon. 2, 371–376 (2008).
[CrossRef]

O. Cohen, H. Buljan, T. Schwartz, J. Fleischer, and M. Segev, “Incoherent solitons in instantaneous nonlocal nonlinear media,” Phys. Rev. E 73, 015601 (2006).
[CrossRef]

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Observation of random-phase lattice solitons,” Nature (London) 433, 500–503(2005).
[CrossRef]

H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides, “Random-phase solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef]

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media,” Phys. Rev. E 63, 035601 (2001).
[CrossRef]

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[CrossRef]

M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000).
[CrossRef]

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, Z. Chen, and M. Segev, “Theory of incoherent dark solitons,” Phys. Rev. Lett. 80, 5113–5116 (1998).
[CrossRef]

M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature (London) 387, 880–883 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media” Phys. Rev. Lett. 78, 646–649 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media,” Phys. Rev. Lett. 78, 646–649 (1997).
[CrossRef]

M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
[CrossRef]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
[CrossRef]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[CrossRef]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
[CrossRef]

Semenov, V. E.

Sergeev, A. M.

A. G. Litvak and A. M. Sergeev, “One dimensional collapse of plasma waves,” JETP Lett. 27, 517–520 (1978).

Serio, M.

M. Onorato, A. Osborne, R. Fedele, and M. Serio, “Landau damping and coherent structures in narrow-banded 1+1 deep water gravity waves,” Phys. Rev. E 67, 046305 (2003).
[CrossRef]

Shih, M.

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[CrossRef]

Shvets, V. F.

V. E. Zakharov, A. N. Pushkarev, V. F. Shvets, and V. V. Yan’kov, “Soliton turbulence,” JETP Lett. 48, 83–87 (1988).

Silberberg, Y.

Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury Brown and Twiss interferometry with interacting photons,” Nat. Photon. 4, 721–726 (2010).
[CrossRef]

Y. Silberberg, Y. Lahini, E. Small, and R. Morandotti, “Universal correlations in a nonlinear periodic 1D system,” Phys. Rev. Lett. 102, 233904 (2009).
[CrossRef]

Skupin, S.

S. Skupin, M. Saffman, and W. Krolikowski, “Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor,” Phys. Rev. Lett. 98, 263902 (2007).
[CrossRef]

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, “Stability of two-dimensional spatial solitons in nonlocal nonlinear media,” Phys. Rev. E 73, 066603 (2006).
[CrossRef]

Small, E.

Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury Brown and Twiss interferometry with interacting photons,” Nat. Photon. 4, 721–726 (2010).
[CrossRef]

Y. Silberberg, Y. Lahini, E. Small, and R. Morandotti, “Universal correlations in a nonlinear periodic 1D system,” Phys. Rev. Lett. 102, 233904 (2009).
[CrossRef]

Snyder, A.

A. Snyder and D. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
[CrossRef]

Snyder, A. W.

N. Akhmediev, W. Krolikowski, and A. W. Snyder, “Partially voherent solitons of variable shape,” Phys. Rev. Lett. 81, 4632–4635 (1998).
[CrossRef]

Soh, D. B. S.

Soljacic, M.

M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000).
[CrossRef]

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[CrossRef]

Stabinis, A.

A. Stabinis, V. Pyragaite, G. Tamoauskas, and A. Piskarskas, “Spectrum of second-harmonic radiation generated from incoherent light,” Phys. Rev. A 84, 043813 (2011).
[CrossRef]

A. Piskarskas, V. Pyragaite, and A. Stabinis, “Generation of coherent waves by frequency up-conversion and down-conversion of incoherent light,” Phys. Rev. A 82, 053817 (2010).
[CrossRef]

Strömqvist, G.

Suret, P.

P. Suret, A. Picozzi, and S. Randoux, “Wave turbulence in integrable systems: nonlinear propagation of incoherent optical waves in single-mode fibers,” Opt. Express 19, 17852–17863 (2011).
[CrossRef]

C. Michel, M. Haelterman, P. Suret, S. Randoux, R. Kaiser, and A. Picozzi, “Thermalization and condensation in an incoherently pumped passive optical cavity,” Phys. Rev. A 84, 033848 (2011).
[CrossRef]

P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, “Anomalous thermalization of nonlinear wave systems,” Phys. Rev. Lett. 104, 054101 (2010).
[CrossRef]

C. Michel, P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, “Influence of third-order dispersion on the propagation of incoherent light in optical fibers,” Opt. Lett. 35, 2367–2369 (2010).
[CrossRef]

Syunyaev, R. A.

Y. B. Zel’dovich, E. V. Levich, and R. A. Syunyaev, “Stimulated Compton interaction between Maxwellian electrons and spectrally narrow radiation,” Sov. Phys. JETP 35, 733–740 (1972).

Tamoauskas, G.

A. Stabinis, V. Pyragaite, G. Tamoauskas, and A. Piskarskas, “Spectrum of second-harmonic radiation generated from incoherent light,” Phys. Rev. A 84, 043813 (2011).
[CrossRef]

Trillo, S.

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef]

N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, “Shocks in nonlocal media,” Phys. Rev. Lett. 99, 043903 (2007).
[CrossRef]

Turitsyn, S. K.

S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castanon, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nat. Photon. 4, 231–235 (2010).
[CrossRef]

E. G. Turitsyna, G. Falkovich, V. K. Mezentsev, and S. K. Turitsyn, “Optical turbulence and spectral condensate in long-fiber lasers,” Phys. Rev. A 80, 031804 (2009).
[CrossRef]

S. A. Babin, V. Karalekas, E. V. Podivilov, V. K. Mezentsev, P. Harper, J. D. Ania-Castanon, and S. K. Turitsyn, “Turbulent broadening of optical spectra in ultralong Raman fiber lasers,” Phys. Rev. A 77, 033803 (2008).
[CrossRef]

Turitsyna, E. G.

E. G. Turitsyna, G. Falkovich, V. K. Mezentsev, and S. K. Turitsyn, “Optical turbulence and spectral condensate in long-fiber lasers,” Phys. Rev. A 80, 031804 (2009).
[CrossRef]

Vewinger, F.

J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468, 545–548 (2010).
[CrossRef]

Vishwanath, A.

M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000).
[CrossRef]

Wan, W.

C. Barsi, W. Wan, and J. W. Fleischer, “Imaging through nonlinear media via digital holography,” Nat. Photon. 3, 211–215 (2009).
[CrossRef]

Weill, R.

Weitz, M.

J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468, 545–548 (2010).
[CrossRef]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Wu, M.

M. Wu, P. Krivosik, B. A. Kalinikos, and C. E. Patton, “Random generation of coherent solitary waves from incoherent waves,” Phys. Rev. Lett. 96, 227202 (2006).
[CrossRef]

Wyller, J.

W. Krolikowski, O. Bang, and J. Wyller, “Nonlocal incoherent solitons,” Phys. Rev. E 70, 036617 (2004).
[CrossRef]

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[CrossRef]

J. Wyller, W. Krolikowski, O. Bang, and J. J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66, 066615 (2002).
[CrossRef]

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619 (2002).
[CrossRef]

Yan’kov, V. V.

V. E. Zakharov, A. N. Pushkarev, V. F. Shvets, and V. V. Yan’kov, “Soliton turbulence,” JETP Lett. 48, 83–87 (1988).

Yariv, A.

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov, F. Dias, and A. Pushkarev, “One-dimensional wave turbulence,” Phys. Rep. 398, 1–65 (2004).
[CrossRef]

S. L. Musher, A. M. Rubenchik, and V. E. Zakharov, “Weak Langmuir turbulence,” Phys. Rep. 252, 177–274 (1995).
[CrossRef]

V. E. Zakharov, A. N. Pushkarev, V. F. Shvets, and V. V. Yan’kov, “Soliton turbulence,” JETP Lett. 48, 83–87 (1988).

V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, “Hamiltonian approach to the description of non-linear plasma phenomena,” Phys. Rep. 129, 285–366 (1985).
[CrossRef]

V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, “Weak Langmuir turbulence of an isothermal plasma,” Sov. Phys. JETP 42, 80–86 (1975).

V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I (Springer, 1992).

Zel’dovich, Y. B.

Y. B. Zel’dovich, E. V. Levich, and R. A. Syunyaev, “Stimulated Compton interaction between Maxwellian electrons and spectrally narrow radiation,” Sov. Phys. JETP 35, 733–740 (1972).

Ann. Rev. Fluids Mech. (1)

A. C. Newell and B. Rumpf, “Wave turbulence,” Ann. Rev. Fluids Mech. 43, 59–78 (2001).

Europhys. Lett. (2)

A. Picozzi and S. Rica, “Coherence absorption and condensation induced by thermalization of incoherent nonlinear fields,” Europhys. Lett. 84, 34004 (2008).
[CrossRef]

S. Lagrange, H. R. Jauslin, and A. Picozzi, “Thermalization of the dispersive three-wave interaction,” Europhys. Lett. 79, 64001 (2007).
[CrossRef]

J. Fluid Mech. (2)

K. Hasselmann, “On the non-linear energy transfer in a gravity-wave spectrum. Part 1. General theory,” J. Fluid Mech. 12, 481–500 (1962).
[CrossRef]

K. Hasselmann, “On the non-linear energy transfer in a gravity-wave spectrum. Part 2. Conservation theorems; wave-particle analogy; irreversibility,” J. Fluid Mech. 15, 273–281 (1963).
[CrossRef]

J. Opt. B (1)

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[CrossRef]

J. Opt. Soc. Am. B (5)

JETP Lett. (2)

A. G. Litvak and A. M. Sergeev, “One dimensional collapse of plasma waves,” JETP Lett. 27, 517–520 (1978).

V. E. Zakharov, A. N. Pushkarev, V. F. Shvets, and V. V. Yan’kov, “Soliton turbulence,” JETP Lett. 48, 83–87 (1988).

Nat. Photon. (4)

C. Rotschild, T. Schwartz, O. Cohen, and M. Segev, “Incoherent spatial solitons in effectively-instantaneous nonlocal nonlinear media,” Nat. Photon. 2, 371–376 (2008).
[CrossRef]

C. Barsi, W. Wan, and J. W. Fleischer, “Imaging through nonlinear media via digital holography,” Nat. Photon. 3, 211–215 (2009).
[CrossRef]

S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castanon, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nat. Photon. 4, 231–235 (2010).
[CrossRef]

Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury Brown and Twiss interferometry with interacting photons,” Nat. Photon. 4, 721–726 (2010).
[CrossRef]

Nature (1)

J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation of photons in an optical microcavity,” Nature 468, 545–548 (2010).
[CrossRef]

Nature (London) (2)

M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature (London) 387, 880–883 (1997).
[CrossRef]

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Observation of random-phase lattice solitons,” Nature (London) 433, 500–503(2005).
[CrossRef]

Opt. Commun. (1)

M. Lisak, L. Helczynski, and D. Anderson, “Relation between different formalisms describing partially incoherent wave propagation in nonlinear optical media,” Opt. Commun. 220, 321–323 (2003).
[CrossRef]

Opt. Express (5)

Opt. Lett. (6)

Phys. Fluids (3)

A. Hasegawa, “Dynamics of an ensemble of plane waves in nonlinear dispersive media,” Phys. Fluids 18, 77–79 (1975).
[CrossRef]

A. Hasegawa, “Envelope soliton of random phase waves,” Phys. Fluids 20, 2155–2156 (1977).
[CrossRef]

C. Montes, J. Peyraud, and M. Hénon, “One-dimensional boson soliton collisions,” Phys. Fluids 22, 176–182 (1979).
[CrossRef]

Phys. Lett. A (1)

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374, 3585–3589 (2010).
[CrossRef]

Phys. Rep. (4)

V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, “Hamiltonian approach to the description of non-linear plasma phenomena,” Phys. Rep. 129, 285–366 (1985).
[CrossRef]

A. Campa, T. Dauxois, and S. Ruffo, “Statistical mechanics and dynamics of solvable models with long-range interactions,” Phys. Rep. 480, 57–159 (2009).
[CrossRef]

S. L. Musher, A. M. Rubenchik, and V. E. Zakharov, “Weak Langmuir turbulence,” Phys. Rep. 252, 177–274 (1995).
[CrossRef]

V. E. Zakharov, F. Dias, and A. Pushkarev, “One-dimensional wave turbulence,” Phys. Rep. 398, 1–65 (2004).
[CrossRef]

Phys. Rev. (1)

I. B. Bernstein, J. M. Green, and M. D. Kruskal, “Exact nonlinear plasma oscillations,” Phys. Rev. 108, 546–550 (1957).
[CrossRef]

Phys. Rev. A (10)

C. Michel, B. Kibler, and A. Picozzi, “Discrete spectral incoherent solitons in nonlinear media with noninstantaneous response,” Phys. Rev. A 83, 023806 (2011).
[CrossRef]

C. Montes, “Photon soliton and fine structure due to nonlinear Compton scattering,” Phys. Rev. A 20, 1081–1095 (1979).
[CrossRef]

J. Garnier and A. Picozzi, “Unified kinetic formulation of incoherent waves propagating in nonlinear media with noninstantaneous response,” Phys. Rev. A 81, 033831 (2010).
[CrossRef]

P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi, “Condensation and thermalization of classsical optical waves in a waveguide,” Phys. Rev. A 83, 033838 (2011).
[CrossRef]

B. Barviau, B. Kibler, and A. Picozzi, “Wave-turbulence approach of supercontinuum generation: influence of self-steepening and higher-order dispersion,” Phys. Rev. A 79, 063840 (2009).
[CrossRef]

A. Piskarskas, V. Pyragaite, and A. Stabinis, “Generation of coherent waves by frequency up-conversion and down-conversion of incoherent light,” Phys. Rev. A 82, 053817 (2010).
[CrossRef]

A. Stabinis, V. Pyragaite, G. Tamoauskas, and A. Piskarskas, “Spectrum of second-harmonic radiation generated from incoherent light,” Phys. Rev. A 84, 043813 (2011).
[CrossRef]

E. G. Turitsyna, G. Falkovich, V. K. Mezentsev, and S. K. Turitsyn, “Optical turbulence and spectral condensate in long-fiber lasers,” Phys. Rev. A 80, 031804 (2009).
[CrossRef]

C. Michel, M. Haelterman, P. Suret, S. Randoux, R. Kaiser, and A. Picozzi, “Thermalization and condensation in an incoherently pumped passive optical cavity,” Phys. Rev. A 84, 033848 (2011).
[CrossRef]

S. A. Babin, V. Karalekas, E. V. Podivilov, V. K. Mezentsev, P. Harper, J. D. Ania-Castanon, and S. K. Turitsyn, “Turbulent broadening of optical spectra in ultralong Raman fiber lasers,” Phys. Rev. A 77, 033803 (2008).
[CrossRef]

Phys. Rev. E (11)

O. Cohen, H. Buljan, T. Schwartz, J. Fleischer, and M. Segev, “Incoherent solitons in instantaneous nonlocal nonlinear media,” Phys. Rev. E 73, 015601 (2006).
[CrossRef]

J. Wyller, W. Krolikowski, O. Bang, and J. J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66, 066615 (2002).
[CrossRef]

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619 (2002).
[CrossRef]

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, “Stability of two-dimensional spatial solitons in nonlocal nonlinear media,” Phys. Rev. E 73, 066603 (2006).
[CrossRef]

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in photonic crystal fiber,” Phys. Rev. E 84, 066605 (2011).
[CrossRef]

M. Onorato, A. Osborne, R. Fedele, and M. Serio, “Landau damping and coherent structures in narrow-banded 1+1 deep water gravity waves,” Phys. Rev. E 67, 046305 (2003).
[CrossRef]

R. Jordan and C. Josserand, “Self-organization in nonlinear wave turbulence,” Phys. Rev. E 61, 1527–1539 (2000).
[CrossRef]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002).
[CrossRef]

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media,” Phys. Rev. E 63, 035601 (2001).
[CrossRef]

A. Picozzi and P. Aschieri, “Influence of dispersion on the resonant interaction between three incoherent waves,” Phys. Rev. E 72, 046606 (2005).
[CrossRef]

W. Krolikowski, O. Bang, and J. Wyller, “Nonlocal incoherent solitons,” Phys. Rev. E 70, 036617 (2004).
[CrossRef]

Phys. Rev. Lett. (35)

D. N. Christodoulides, T. H. Coskun, M. Mitchell, Z. Chen, and M. Segev, “Theory of incoherent dark solitons,” Phys. Rev. Lett. 80, 5113–5116 (1998).
[CrossRef]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media” Phys. Rev. Lett. 78, 646–649 (1997).
[CrossRef]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
[CrossRef]

O. Bang, D. Edmundson, and W. Krolikowski, “Collapse of incoherent light beams in inertial bulk Kerr media,” Phys. Rev. Lett. 83, 5479–5482 (1999).
[CrossRef]

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity locking of incoherent nonlinear wave packets,” Phys. Rev. Lett. 97, 033902 (2006).
[CrossRef]

N. Akhmediev, W. Krolikowski, and A. W. Snyder, “Partially voherent solitons of variable shape,” Phys. Rev. Lett. 81, 4632–4635 (1998).
[CrossRef]

T. Hansson, M. Lisak, and D. Anderson, “Integrability and conservation laws for the nonlinear evolution equations of partially coherent waves in noninstantaneous Kerr media,” Phys. Rev. Lett. 108, 063901 (2012).
[CrossRef]

M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990–4993 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media,” Phys. Rev. Lett. 78, 646–649 (1997).
[CrossRef]

H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides, “Random-phase solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef]

D. V. Dylov and J. W. Fleischer, “Observation of all-optical bump-on-tail instability,” Phys. Rev. Lett. 100, 103903 (2008).
[CrossRef]

P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, “Anomalous thermalization of nonlinear wave systems,” Phys. Rev. Lett. 104, 054101 (2010).
[CrossRef]

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

B. Rumpf and A. C. Newell, “Coherent structures and entropy in constrained, modulationally unstable, nonintegrable systems,” Phys. Rev. Lett. 87, 054102 (2001).
[CrossRef]

A. Picozzi, S. Pitois, and G. Millot, “Spectral incoherent solitons: a localized soliton behavior in the frequency domain,” Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef]

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, “Observation of attraction between dark solitons,” Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef]

A. Picozzi and J. Garnier, “Incoherent soliton turbulence in nonlocal nonlinear media,” Phys. Rev. Lett. 107, 233901(2011).
[CrossRef]

A. Picozzi and M. Haelterman, “Parametric three-wave soliton generated from incoherent light,” Phys. Rev. Lett. 86, 2010–2013 (2001).
[CrossRef]

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, “Incoherent solitons in instantaneous response nonlinear media,” Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef]

M. Wu, P. Krivosik, B. A. Kalinikos, and C. E. Patton, “Random generation of coherent solitary waves from incoherent waves,” Phys. Rev. Lett. 96, 227202 (2006).
[CrossRef]

S. Skupin, M. Saffman, and W. Krolikowski, “Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor,” Phys. Rev. Lett. 98, 263902 (2007).
[CrossRef]

C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef]

C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
[CrossRef]

N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, “Shocks in nonlocal media,” Phys. Rev. Lett. 99, 043903 (2007).
[CrossRef]

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef]

A. Picozzi and M. Haelterman, “Hidden coherence along space-time trajectories in parametric wave mixing,” Phys. Rev. Lett. 88, 083901 (2002).
[CrossRef]

Y. Silberberg, Y. Lahini, E. Small, and R. Morandotti, “Universal correlations in a nonlinear periodic 1D system,” Phys. Rev. Lett. 102, 233904 (2009).
[CrossRef]

C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, “Highly noninstantaneous solitons in liquid-core photonic crystal fibers,” Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

A. Picozzi and M. Haelterman, “Condensation in Hamiltonian parametric wave interaction,” Phys. Rev. Lett. 92, 103901 (2004).
[CrossRef]

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, “Condensation in disordered lasers: theory, 3D+1 simulations, and experiments,” Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

R. Weill, B. Fischer, and O. Gat, “Light-mode condensation in actively-mode-locked lasers,” Phys. Rev. Lett. 104, 173901 (2010).
[CrossRef]

M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation instability of incoherent beams in noninstantaneous nonlinear media,” Phys. Rev. Lett. 84, 467–470 (2000).
[CrossRef]

M. J. Davis, S. A. Morgan, and K. Burnett, “Simulations of Bose fields at finite temperature,” Phys. Rev. Lett. 87, 160402 (2001).
[CrossRef]

Physica D (3)

B. Rumpf and A. C. Newell, “Localization and coherence in nonintegrable systems,” Physica D 184, 162–191 (2003).
[CrossRef]

G. Düring, A. Picozzi, and S. Rica, “Breakdown of weak-turbulence and nonlinear wave condensation,” Physica D 238, 1524–1549 (2009).
[CrossRef]

A. C. Newell, S. Nazarenko, and L. Biven, “Wave turbulence and intermittency,” Physica D 152, 520–550 (2001).
[CrossRef]

Rep. Prog. Phys. (1)

T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, “The physics of dipolar bosonic quantum gases,” Rep. Prog. Phys. 72, 126401 (2009).
[CrossRef]

Rev. Geophys. (1)

A. C. Newell, “The closure problem in a system of random gravity waves,” Rev. Geophys. 6, 1–31 (1968).
[CrossRef]

Science (3)

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998).
[CrossRef]

A. Snyder and D. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
[CrossRef]

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000).
[CrossRef]

Sov. Phys. JETP (3)

Y. B. Zel’dovich, E. V. Levich, and R. A. Syunyaev, “Stimulated Compton interaction between Maxwellian electrons and spectrally narrow radiation,” Sov. Phys. JETP 35, 733–740 (1972).

V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, “Weak Langmuir turbulence of an isothermal plasma,” Sov. Phys. JETP 42, 80–86 (1975).

G. A. Pasmanik, “Self-interaction of incoherent light beams,” Sov. Phys. JETP 39, 234–238 (1974).

Other (5)

S. Nazarenko, Wave Turbulence, Lectures Notes in Physics825 (Springer, 2011).

R. W. Boyd, Nonlinear Optics (Academic Press, 2008).

Y. S. Kivshar and G. P. Agrawal, Optical Solitons : From Fibers to Photonic Crystals (Academic Press, 2003).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I (Springer, 1992).

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Figures (3)

Fig. 1.
Fig. 1.

Schematic illustration of the validity of the fundamental kinetic equations in the framework of a spatially nonlocal nonlinear response: σs/Λs represents the amount of nonlocality of the nonlinear interaction, while λc/Δs represents the amount of inhomogeneous statistics of the incoherent wave. When the incoherent wave is characterized by fluctuations that are statistically homogeneous in space, the relevant kinetic description is provided by the WT (Hasselmann) kinetic equation, which describes in particular the irreversible process of thermalization to the thermodynamic Rayleigh–Jeans equilibrium spectrum. When the incoherent wave exhibits an inhomogeneous statistics, the relevant kinetic description is provided by different variants of the Vlasov equation, whose self-consistent potential depends on the amount of nonlocality in the system. The Vlasov kinetic equation describes, in particular, incoherent modulational instability and localized IS structures. (σs is the range of the nonlocal interaction, Λs is the healing length, λc is the correlation length, and Δs is the length scale of inhomogeneous statistics).

Fig. 2.
Fig. 2.

Schematic illustration of the validity of the fundamental kinetic equations in the framework of a temporally noninstantaneous nonlinear response: σt/Λt represents the amount of noninstantaneous response of the nonlinearity, while tc/Δt represents the amount of nonstationary statistics of the incoherent wave. The diagram for the temporal domain reported here is similar to that reported in the spatial domain in Fig. 1. The essential difference between the spatial and the temporal domain relies on the fact that in the temporal domain, the response function is constrained by the causality condition. It turns out that when the finite response time of the nonlinearity cannot be neglected, the relevant kinetic description is provided by an equation analogous to the weak Langmuir turbulence equation, irrespective of the nature of the fluctuations that may be either stationary or nonstationary. This equation has been shown to describe nonlocalized spectral ISs. In the presence of a highly noninstantaneous nonlinear response and a stationary statistics of the incoherent wave, the weak Langmuir turbulence reduces to the Korteweg–de Vries equation. Conversely, when the wave exhibits a nonstationary statistics still in the presence of a highly noninstantaneous response, we derive a “temporal long-range” Vlasov equation, whose self-consistent potential is constrained by the causality condition of the noninstantaneous response function. (σt is the response time of the nonlinearity, Λt is the “healing time,” tc is the correlation time, and Δt is the time scale of nonstationary statistics).

Fig. 3.
Fig. 3.

Analogy between a system of classical particles and the propagation of an incoherent optical wave in a cubic nonlinear medium. (a) As described by the kinetic gas theory (Boltzmann kinetic equation), collisions between particles are responsible for an irreversible evolution of the gas towards thermodynamic equilibrium. (b) In complete analogy, the WT (Hasselmann) kinetic equation and the underlying four-wave mixing describe an irreversible evolution of the incoherent optical wave toward the thermodynamic Rayleigh–Jeans equilibrium state. (c) When the incoherent optical wave exhibits an inhomogeneous statistics, the four-wave interaction no longer takes place locally; i.e., the quasi-particles feel the presence of an effective self-consistent potential, V(r), which prevents them from relaxing to thermal equilibrium. The dynamics of the incoherent optical wave turns out to be described by a Vlasov-like kinetic equation. (d) In the presence of a noninstantaneous nonlinear interaction, the causality condition inherent to the response function changes the physical picture: the nonlinear interaction involves a material excitation (e.g., molecular vibration in the example of Raman scattering). The dynamics of the incoherent optical wave turns out to be described by a kinetic equation analogous to the weak Langmuir turbulence equation. Note, however, that a highly noninstantaneous nonlinear response is no longer described by the weak Langmuir turbulence equation, but instead by the “long-range” Vlasov-like equation (see Fig. 2).

Equations (53)

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izψ=βs2ψ+γψU(xx)|ψ|2(z,x)dx,
izB(x,ξ,z)=2βsx·ξB(x,ξ,z)+γP(x,ξ,z)+γQ(x,ξ,z),
P(x,ξ)=B(x,ξ)U(y)[N(xy+ξ/2)N(xyξ/2)]dy,
Q(x,ξ)=U(y)[B(xy/2+ξ/2,y)B(xy/2,ξy)B(xy/2,ξ+y)B(xy/2ξ/2,y)]dy,
N(x,z)B(x,ξ=0,z)=|ψ(x,z)|2
nk(x,z)=B(x,ξ,z)exp(ik·ξ)dξ,
B(x,ξ,z)=B(0)(εx,ξ,εz)+O(ε),
znk(x,z)+kω˜k(x,z)·xnk(x,z)xω˜k(x,z)·knk(x,z)=0.
ω˜k(x,z)=ω(k)+Vk(x,z),
Vk(x,z)=γ(2π)d(1+U˜kk)nk(x,z)dk,
N(x,z)=1(2π)dnk(x,z)dk
V(x,z)=2γN(x,z).
U(x)=εU(0)(εx).
V(x,z)=γU(xx)N(x,z)dx.
V(x)=γNU(x).
izψ=βs2ψ+γNU(x)ψ.
izψ=βtttψ+γψ+R(tt)|ψ|2(z,t)dt,
R˜(ω)=U˜(ω)+ig(ω),
nω(t,z)=+B(t,τ,z)exp(iωτ)dτ,
B(t,τ,z)=B(0)(εt,τ,εz)+O(ε),
znω(t,z)=γπnω(t,z)+g(ωω)nω(t,z)dω,
znω(z)=γπnω(z)+g(ωω)nω(z)dω.
nω(z)=n0+n˜ω(z),
zn˜ω(z)γn0g1πωn˜ω(z)=γg1πn˜ω(z)ωn˜ω(z)+γn0g36πω3n˜ω(z),
g1=+ωg(ω)dω,g3=+ω3g(ω)dω.
znω(t,z)+ωk˜ω(t,z)tnω(t,z)tk˜ω(t,z)ωnω(t,z)=0,
k˜ω(t,z)=k(ω)+V(t,z),
V(t,z)=γR(tt)N(t,z)dt.
U(t)=12πU˜(ω)exp(iωt)dω,
G(t)=i2πg(ω)exp(iωt)dω.
znω(t,z)+ωkω(t,z)tnω(t,z)tVU(t,z)ωnω(t,z)=tVG(t,z)ωnω(t,z),
V(t,z)=2γN(t,z).
V(t)=γNU(t).
znk=4πγ2(2π)2dQ(nk,nk1,nk2,nk3)Tk1232δ(k1+k2k3k)δ(βs(k12+k22k32k2))dk1dk2dk3,
nkRJ=Tβsk2μ.
znk=4πγ2ε2d2(2π)2dQ(nk,nk+εκ1,nk+εκ2,nk+εκ1+εκ2)[U˜(0)(κ1κ2)+U˜(0)(κ1+κ2)+2U˜(0)(κ2)]×δ(2βsκ1·κ2)dκ1dκ2.
λknonlocλkloc(σs/Λs)2d2.
P(x,ξ,z)exp(ik·ξ)dξ=1(2π)2dU(y)[nk2(0)(Xεy+εξ/2,Z)nk2(0)(Xεyεξ/2,Z)]×nk1(0)(X,Z)exp[i(k1k)·ξ]dk1dk2dξdy.
P(x,ξ,z)exp(ik·ξ)dξ=iεknk(0)(X,Z)·X(1(2π)dnk(0)(X,Z)dk).
Q˜0=1(2π)dU(y)nk1(0)(X,Z)nk(0)(X,Z)[exp[i(k1k)·y]exp[i(k1k)·y]]dk1dy.
Q˜0=2i(2π)dnk(0)(X,Z)(U˜kk)nk(0)(X,Z)dk,
Q˜1=1(2π)2dU(y)[(u++v)exp[i(k1k2)·y]+(uv)exp[i(k1k2)·y]]exp[i(kk2)·ξ]dk1dk2dydξ,
Q˜1=i2(2π)dX·U(y)k1(nk1(0)(X,Z))nk(0)(X,Z)[exp[i(k1k)·y]+exp[i(k1k)·y]]dk1dy+i2(2π)dk·U(y)nk(0)(X,Z)X(nk1(0)(X,Z))[exp[i(k1k)·y]+exp[i(k1k)·y]]dk1dy
Q˜1=i(2π)dX(U˜kk1nk1(0)(X,Z)dk1)·knk(0)(X,Z)i(2π)dk(U˜kk1nk1(0)(X,Z)dk1)·Xnk(0)(X,Z).
P(x,ξ,z)exp(ik·ξ)dξ=1(2π)dU(0)(Y)[nk2(0)(XY+εξ/2,Z)nk2(0)(XYεξ/2,Z)]nk1(0)(X,Z)exp[i(k1k)·ξ]dk1dk2dξdY.
P(x,ξ,z)exp(ik·ξ)dξ=iεknk(0)(X,Z)·XU(0)(XX)(1(2π)dnk(0)(X,Z)dk)dX.
+P(t,τ,z)exp(iωτ)dτ=iεωnω(0)(T,Z)×T(12π+nω(0)(T,Z)dω).
Q˜0=iπnω(0)(T,Z)+(R˜ωω)nω(0)(T,Z)dω,
γπnω+g(ωω)nωdω=γεπ[n0+ε2n˜ω(0)]+g(0)(ω)[n0+ε2n˜ωεω(0)]dω=γε4π[n0+ε2n˜ω(0)]+g(0)(ω)[ωωn˜ω(0)ω3ε26ω3n˜ω(0)]dω=γε4n0g1(0)πωn˜ω(0)+γε6g1(0)πn˜ω(0)ωn˜ω(0)+γε6n0g3(0)6πω3n˜ω(0),
g1(0)=+ωg(0)(ω)dω,g3(0)=+ω3g(0)(ω)dω.
znω(z)=ε4Zn˜ωε(ε2z),
Zn˜ω(0)(Z)γn0g1(0)πωn˜ω(0)(Z)=γε2g1(0)πn˜ω(0)(Z)×ωn˜ω(0)(Z)+γε2n0g3(0)6πω3n˜ω(0)(Z).
+P(t,τ,z)exp(iωτ)dτ=iεωnω(0)(T,Z)×T+R(0)(TT)(12π+nω(0)(T,Z)dω)dT.

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