Abstract

We study the over-focusing of spatial light beams due to self-focusing nonlinearity, in both local and nonlocal nonlinear media. Numerical simulation of both cases reveals a peaked profile, with a near-cusp at the center surrounded by exponentially-decaying tails, at a critical self-focusing power. The profile is a local effect, occurring as diffraction counteracts nonlinearity. Nonlocality, however, is needed to prevent modulation instability of the initial beam and to prevent catastrophic collapse in 2D. The peaked profile remains for weak nonlocality but disappears for wide nonlocal responses. Beyond the critical power for a peaked solution, or for longer propagation distances, competition between nonlinearity and diffraction causes oscillatory collapse-bounce behavior. The numerical results are confirmed by observing these dynamics in a self-focusing glass with a nonlocal, thermal response.

©2008 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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2008 (1)

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

2007 (6)

2006 (2)

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]

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

2005 (5)

A. I. Yakimenko, Y. A. Zaliznyak, and Y. S. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E 71, 065603 (2005).
[Crossref]

M. Ablowitz, I. Bakirtaş, and B. Ilan, “Wave collapse in a class of nonlocal nonlinear Schrödinger equation,” Physica D 207, 230–253 (2005).
[Crossref]

P. D. Rasmussen, O. Bang, and W. Krolikowski, “Theory of nonlocal soliton interaction in nematic liquid crystals,” Phys. Rev. E 72066611 (2005).
[Crossref]

C. Rotschild, O. Cohen, O. Manela, and M. Segev, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett 95, 213904, (2005).
[Crossref] [PubMed]

J. E. Rothenberg and D. Grischkowsky, “Observation of the formation of an optical intensity shock and wave breaking in the nonlinear propagation of pulses in optical fibres,” Phys. Rev. Lett. 94, 040403 (2005).

2004 (1)

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

2003 (3)

K. D. Moll, A. L. Gaeta, and G. Fibich. “Self-similar optical wave collapse: observation of the Townes profile,” Phys. Rev. Lett. 90, 203902, (2003).
[Crossref] [PubMed]

J. J. Garcia-Ripoll, V. V. Konotop, B. Malomed, and V. M. Perez-Garcia, “A quasi-local Gross-Pitaevskii equation for attractive Bose-Einstein condensates,” Math. And Comp. in Sim. 6221–30 (2003).
[Crossref]

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

2002 (2)

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

M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, ““Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. 27, 1460–1462 (2002).
[Crossref]

2000 (1)

V. M. Perez-Garcia, V. V. Konotop, and J. J. Garcia-Ripoll, “Dynamics of quasicollapse in nonlinear Schrödinger equation with nonlocal interactions,” Phys. Rev. E 62, 4300–4308 (2000).
[Crossref]

1999 (3)

G. Fibich and G. Papanicolaou, “Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension,” SIAM J. Appl. Math. 60, 183–240, (1999).
[Crossref]

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

J. C. Bronski and J. N. Kutz, “Numerical simulation of the semi-classical limit of the focusing nonlinear Schrodinger equation,” Phys. Lett. A 254, 325–336 (1999).
[Crossref]

1998 (2)

P. D. Miller and S. Kamvissis, “On the semiclassical limit of the focusing nonlinear Schrodinger equation,” Phys. Lett. A 247, 75–86 (1998).
[Crossref]

L. Berge, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303, 259–370, (1998).
[Crossref]

1997 (1)

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

1996 (1)

G. Fibich, “Small beam nonparaxiality arrests self-focusing of optical beams,” Phys. Rev. Lett. 76, 4356–4359, (1996).
[Crossref] [PubMed]

1995 (1)

G. A. El and A. L. Krylov, “General-solution of the Cauchy-problem for the defocusing NLS equation in the Whitham limit,” Phys. Rev. A 203, 77–82 (1995).

1994 (1)

O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Subcritical localization in the discrete nonlinear Schrödinger-equation with arbitrary power nonlinearity,” Nonlinearity 7, 205–218 (1994).
[Crossref]

1993 (1)

R. Camassa and D. D. Holm, “An integrable shallow water equation with peaked solitons,” Phys. Rev. Lett. 71, 1661–1664 (1993).
[Crossref] [PubMed]

1992 (2)

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

V. N Tsytovich and R Bingham, “Arrest of wave collapse and transitional damping,” Comments Plasma Phys. Controlled Fusion 14361–368 (1992).

1987 (1)

A. V. Gurevich and A. L. Krylov, “Nondissipative shock waves in media with positive dispersion,” Zh. Eksp. Teor. Fiz. 921684–1699 (1987).

1985 (2)

W. J. Tomlinson, R. H. Stolen, and A. M. Johnson, “Optical wave breaking of pulses in nonlinear optical fibers,” Opt. Lett. 10, 457–459 (1985).
[Crossref] [PubMed]

S. K. Turitsyn, “Spatial dispersion of nonlinearity and stability of multidimensional solitons,” Teor. Mat. Fiz. 64, 226–232, (1985).

1978 (1)

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

1968 (1)

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-action of laser beams,” IEEE J. Quantum Electron. QE-4, 568–575 (1968).
[Crossref]

1965 (1)

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8, (1965).
[Crossref]

1964 (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482, (1964).
[Crossref]

1927 (1)

E. Madelung, “Quantetheorie in hydrodynamischer form’” Z. Phys. 40, 322–326 (1927).
[Crossref]

Ablowitz, M.

M. Ablowitz, I. Bakirtaş, and B. Ilan, “Wave collapse in a class of nonlocal nonlinear Schrödinger equation,” Physica D 207, 230–253 (2005).
[Crossref]

Akhmanov, S. A.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-action of laser beams,” IEEE J. Quantum Electron. QE-4, 568–575 (1968).
[Crossref]

Alberucci, A.

Assanto, G.

Bakirtas, I.

M. Ablowitz, I. Bakirtaş, and B. Ilan, “Wave collapse in a class of nonlocal nonlinear Schrödinger equation,” Physica D 207, 230–253 (2005).
[Crossref]

Bang, O.

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]

P. D. Rasmussen, O. Bang, and W. Krolikowski, “Theory of nonlocal soliton interaction in nematic liquid crystals,” Phys. Rev. E 72066611 (2005).
[Crossref]

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

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

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

O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Subcritical localization in the discrete nonlinear Schrödinger-equation with arbitrary power nonlinearity,” Nonlinearity 7, 205–218 (1994).
[Crossref]

Barsi, C.

Berge, L.

L. Berge, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303, 259–370, (1998).
[Crossref]

Bingham, R

V. N Tsytovich and R Bingham, “Arrest of wave collapse and transitional damping,” Comments Plasma Phys. Controlled Fusion 14361–368 (1992).

Bronski, J. C.

J. C. Bronski and J. N. Kutz, “Numerical simulation of the semi-classical limit of the focusing nonlinear Schrodinger equation,” Phys. Lett. A 254, 325–336 (1999).
[Crossref]

Brzdakiewicz, K. A.

Buljan, H.

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

Camassa, R.

R. Camassa and D. D. Holm, “An integrable shallow water equation with peaked solitons,” Phys. Rev. Lett. 71, 1661–1664 (1993).
[Crossref] [PubMed]

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482, (1964).
[Crossref]

Christiansen, P. L.

O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Subcritical localization in the discrete nonlinear Schrödinger-equation with arbitrary power nonlinearity,” Nonlinearity 7, 205–218 (1994).
[Crossref]

Cohen, O.

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

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

C. Rotschild, O. Cohen, O. Manela, and M. Segev, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett 95, 213904, (2005).
[Crossref] [PubMed]

Conti, C.

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

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

A. Fratalocchi, C. Conti, G. Ruocco, and S. Trillo are preparing a manuscript to be called “Thermodynamics of soliton gases: phase transitions and shock waves”.

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo are preparing a manuscript to be called “Observation of a gradient catastrophe generating solitons”.

Crosignani, B.

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

Dresichuh, A.

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, “MI, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B: Quan. Semiclass. Opt. 6, S288–S294 (2004).
[Crossref]

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

El, G. A.

G. A. El and A. L. Krylov, “General-solution of the Cauchy-problem for the defocusing NLS equation in the Whitham limit,” Phys. Rev. A 203, 77–82 (1995).

Fibich, G.

K. D. Moll, A. L. Gaeta, and G. Fibich. “Self-similar optical wave collapse: observation of the Townes profile,” Phys. Rev. Lett. 90, 203902, (2003).
[Crossref] [PubMed]

G. Fibich and G. Papanicolaou, “Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension,” SIAM J. Appl. Math. 60, 183–240, (1999).
[Crossref]

G. Fibich, “Small beam nonparaxiality arrests self-focusing of optical beams,” Phys. Rev. Lett. 76, 4356–4359, (1996).
[Crossref] [PubMed]

Fischer, B.

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

Fleischer, J. W.

W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46–51 (2007).
[Crossref]

C. Barsi, W. Wan, C. Sun, and J. W. Fleischer, “Dispersive shock waves with nonlocal nonlinearity,” Opt. Lett. 32, 2930–2932 (2007).
[Crossref] [PubMed]

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

Fratalocchi, A.

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo are preparing a manuscript to be called “Observation of a gradient catastrophe generating solitons”.

A. Fratalocchi, C. Conti, G. Ruocco, and S. Trillo are preparing a manuscript to be called “Thermodynamics of soliton gases: phase transitions and shock waves”.

Gaeta, A. L.

K. D. Moll, A. L. Gaeta, and G. Fibich. “Self-similar optical wave collapse: observation of the Townes profile,” Phys. Rev. Lett. 90, 203902, (2003).
[Crossref] [PubMed]

Garcia-Ripoll, J. J.

J. J. Garcia-Ripoll, V. V. Konotop, B. Malomed, and V. M. Perez-Garcia, “A quasi-local Gross-Pitaevskii equation for attractive Bose-Einstein condensates,” Math. And Comp. in Sim. 6221–30 (2003).
[Crossref]

V. M. Perez-Garcia, V. V. Konotop, and J. J. Garcia-Ripoll, “Dynamics of quasicollapse in nonlinear Schrödinger equation with nonlocal interactions,” Phys. Rev. E 62, 4300–4308 (2000).
[Crossref]

Garmire, E.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482, (1964).
[Crossref]

Ghofraniha, N.

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

Gordon, J. P.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8, (1965).
[Crossref]

Grischkowsky, D.

J. E. Rothenberg and D. Grischkowsky, “Observation of the formation of an optical intensity shock and wave breaking in the nonlinear propagation of pulses in optical fibres,” Phys. Rev. Lett. 94, 040403 (2005).

Gurevich, A. V.

A. V. Gurevich and A. L. Krylov, “Nondissipative shock waves in media with positive dispersion,” Zh. Eksp. Teor. Fiz. 921684–1699 (1987).

Holm, D. D.

R. Camassa and D. D. Holm, “An integrable shallow water equation with peaked solitons,” Phys. Rev. Lett. 71, 1661–1664 (1993).
[Crossref] [PubMed]

Ilan, B.

M. Ablowitz, I. Bakirtaş, and B. Ilan, “Wave collapse in a class of nonlocal nonlinear Schrödinger equation,” Physica D 207, 230–253 (2005).
[Crossref]

Jia, S.

W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46–51 (2007).
[Crossref]

Johnson, A. M.

Kamchatnov, A. M.

A. M. Kamchatnov, Nonlinear Periodic Waves and Their Modulations (World Scientific, Singapore2000).
[Crossref]

Kaminer, I.

Kamvissis, S.

P. D. Miller and S. Kamvissis, “On the semiclassical limit of the focusing nonlinear Schrodinger equation,” Phys. Lett. A 247, 75–86 (1998).
[Crossref]

Khokhlov, R. V.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-action of laser beams,” IEEE J. Quantum Electron. QE-4, 568–575 (1968).
[Crossref]

Kivshar, Y. S.

A. Minovich, D. N. Neshev, A. Dresichuh, W. Krolikowski, and Y. S. Kivshar, “Experimental reconstruction of nonlocal response of thermal nonlinear optical media,” Opt. Lett. 32, 1599–1601 (2007).
[Crossref] [PubMed]

A. I. Yakimenko, Y. A. Zaliznyak, and Y. S. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E 71, 065603 (2005).
[Crossref]

Konotop, V. V.

J. J. Garcia-Ripoll, V. V. Konotop, B. Malomed, and V. M. Perez-Garcia, “A quasi-local Gross-Pitaevskii equation for attractive Bose-Einstein condensates,” Math. And Comp. in Sim. 6221–30 (2003).
[Crossref]

V. M. Perez-Garcia, V. V. Konotop, and J. J. Garcia-Ripoll, “Dynamics of quasicollapse in nonlinear Schrödinger equation with nonlocal interactions,” Phys. Rev. E 62, 4300–4308 (2000).
[Crossref]

Krindach, D. P.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-action of laser beams,” IEEE J. Quantum Electron. QE-4, 568–575 (1968).
[Crossref]

Krolikowski, W.

A. Minovich, D. N. Neshev, A. Dresichuh, W. Krolikowski, and Y. S. Kivshar, “Experimental reconstruction of nonlocal response of thermal nonlinear optical media,” Opt. Lett. 32, 1599–1601 (2007).
[Crossref] [PubMed]

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]

P. D. Rasmussen, O. Bang, and W. Krolikowski, “Theory of nonlocal soliton interaction in nematic liquid crystals,” Phys. Rev. E 72066611 (2005).
[Crossref]

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

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

Królikowski, W.

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

Krylov, A. L.

G. A. El and A. L. Krylov, “General-solution of the Cauchy-problem for the defocusing NLS equation in the Whitham limit,” Phys. Rev. A 203, 77–82 (1995).

A. V. Gurevich and A. L. Krylov, “Nondissipative shock waves in media with positive dispersion,” Zh. Eksp. Teor. Fiz. 921684–1699 (1987).

Kutz, J. N.

J. C. Bronski and J. N. Kutz, “Numerical simulation of the semi-classical limit of the focusing nonlinear Schrodinger equation,” Phys. Lett. A 254, 325–336 (1999).
[Crossref]

Leite, R. C. C.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8, (1965).
[Crossref]

Litvak, A. G.

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

Madelung, E.

E. Madelung, “Quantetheorie in hydrodynamischer form’” Z. Phys. 40, 322–326 (1927).
[Crossref]

Malomed, B.

J. J. Garcia-Ripoll, V. V. Konotop, B. Malomed, and V. M. Perez-Garcia, “A quasi-local Gross-Pitaevskii equation for attractive Bose-Einstein condensates,” Math. And Comp. in Sim. 6221–30 (2003).
[Crossref]

Manela, O.

I. Kaminer, C. Rotschild, O. Manela, and M. Segev, “Periodic solitons in nonlocal nonlinear Media,” Opt. Lett. 32, 3209–3211, (2007).
[Crossref] [PubMed]

C. Rotschild, O. Cohen, O. Manela, and M. Segev, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett 95, 213904, (2005).
[Crossref] [PubMed]

Migulin, A. V.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-action of laser beams,” IEEE J. Quantum Electron. QE-4, 568–575 (1968).
[Crossref]

Miller, P. D.

P. D. Miller and S. Kamvissis, “On the semiclassical limit of the focusing nonlinear Schrodinger equation,” Phys. Lett. A 247, 75–86 (1998).
[Crossref]

Minovich, A.

Mitchell, D. J.

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

Moll, K. D.

K. D. Moll, A. L. Gaeta, and G. Fibich. “Self-similar optical wave collapse: observation of the Townes profile,” Phys. Rev. Lett. 90, 203902, (2003).
[Crossref] [PubMed]

Moore, R. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8, (1965).
[Crossref]

Neshev, D.

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

Neshev, D. N.

Nikolov, N. I.

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

Papanicolaou, G.

G. Fibich and G. Papanicolaou, “Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension,” SIAM J. Appl. Math. 60, 183–240, (1999).
[Crossref]

Peccianti, M.

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

M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, ““Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. 27, 1460–1462 (2002).
[Crossref]

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo are preparing a manuscript to be called “Observation of a gradient catastrophe generating solitons”.

Perez-Garcia, V. M.

J. J. Garcia-Ripoll, V. V. Konotop, B. Malomed, and V. M. Perez-Garcia, “A quasi-local Gross-Pitaevskii equation for attractive Bose-Einstein condensates,” Math. And Comp. in Sim. 6221–30 (2003).
[Crossref]

V. M. Perez-Garcia, V. V. Konotop, and J. J. Garcia-Ripoll, “Dynamics of quasicollapse in nonlinear Schrödinger equation with nonlocal interactions,” Phys. Rev. E 62, 4300–4308 (2000).
[Crossref]

Porto, S. P. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8, (1965).
[Crossref]

Rasmussen, J.

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

Rasmussen, J. J.

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

O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Subcritical localization in the discrete nonlinear Schrödinger-equation with arbitrary power nonlinearity,” Nonlinearity 7, 205–218 (1994).
[Crossref]

Rasmussen, P. D.

P. D. Rasmussen, O. Bang, and W. Krolikowski, “Theory of nonlocal soliton interaction in nematic liquid crystals,” Phys. Rev. E 72066611 (2005).
[Crossref]

Rothenberg, J. E.

J. E. Rothenberg and D. Grischkowsky, “Observation of the formation of an optical intensity shock and wave breaking in the nonlinear propagation of pulses in optical fibres,” Phys. Rev. Lett. 94, 040403 (2005).

Rotschild, C.

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

I. Kaminer, C. Rotschild, O. Manela, and M. Segev, “Periodic solitons in nonlocal nonlinear Media,” Opt. Lett. 32, 3209–3211, (2007).
[Crossref] [PubMed]

C. Rotschild, O. Cohen, O. Manela, and M. Segev, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett 95, 213904, (2005).
[Crossref] [PubMed]

Ruocco, G.

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

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo are preparing a manuscript to be called “Observation of a gradient catastrophe generating solitons”.

A. Fratalocchi, C. Conti, G. Ruocco, and S. Trillo are preparing a manuscript to be called “Thermodynamics of soliton gases: phase transitions and shock waves”.

Schwartz, T.

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

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

Segev, M.

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

I. Kaminer, C. Rotschild, O. Manela, and M. Segev, “Periodic solitons in nonlocal nonlinear Media,” Opt. Lett. 32, 3209–3211, (2007).
[Crossref] [PubMed]

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

C. Rotschild, O. Cohen, O. Manela, and M. Segev, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett 95, 213904, (2005).
[Crossref] [PubMed]

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

Sergeev, A. M.

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

Skupin, S.

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]

Snyder, A. W.

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

Stolen, R. H.

Sukhorukov, A. P.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-action of laser beams,” IEEE J. Quantum Electron. QE-4, 568–575 (1968).
[Crossref]

Sulem, C.

C. Sulem and P. L. Sulem., The nonlinear Schrodinger equation: self-focusing and wave collapse (Springer-Verlag, New York, 1999).

Sulem, P. L.

C. Sulem and P. L. Sulem., The nonlinear Schrodinger equation: self-focusing and wave collapse (Springer-Verlag, New York, 1999).

Sun, C.

Tomlinson, W. J.

Townes, C. H.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482, (1964).
[Crossref]

Trillo, S.

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

A. Fratalocchi, C. Conti, G. Ruocco, and S. Trillo are preparing a manuscript to be called “Thermodynamics of soliton gases: phase transitions and shock waves”.

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo are preparing a manuscript to be called “Observation of a gradient catastrophe generating solitons”.

Tsytovich, V. N

V. N Tsytovich and R Bingham, “Arrest of wave collapse and transitional damping,” Comments Plasma Phys. Controlled Fusion 14361–368 (1992).

Turitsyn, S. K.

S. K. Turitsyn, “Spatial dispersion of nonlinearity and stability of multidimensional solitons,” Teor. Mat. Fiz. 64, 226–232, (1985).

Wan, W.

C. Barsi, W. Wan, C. Sun, and J. W. Fleischer, “Dispersive shock waves with nonlocal nonlinearity,” Opt. Lett. 32, 2930–2932 (2007).
[Crossref] [PubMed]

W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46–51 (2007).
[Crossref]

Whinnery, J. R.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8, (1965).
[Crossref]

Wyller, J.

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

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

Yakimenko, A. I.

A. I. Yakimenko, Y. A. Zaliznyak, and Y. S. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E 71, 065603 (2005).
[Crossref]

Yariv, A.

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

Zaliznyak, Y. A.

A. I. Yakimenko, Y. A. Zaliznyak, and Y. S. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E 71, 065603 (2005).
[Crossref]

Comments Plasma Phys. Controlled Fusion (1)

V. N Tsytovich and R Bingham, “Arrest of wave collapse and transitional damping,” Comments Plasma Phys. Controlled Fusion 14361–368 (1992).

IEEE J. Quantum Electron. (1)

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, and R. V. Khokhlov, “Thermal self-action of laser beams,” IEEE J. Quantum Electron. QE-4, 568–575 (1968).
[Crossref]

J. Appl. Phys. (1)

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8, (1965).
[Crossref]

J. Opt. B: Quan. Semiclass. Opt. (1)

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

J. Opt. Soc. Am. B (1)

JETP Lett. (1)

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

Math. And Comp. in Sim. (1)

J. J. Garcia-Ripoll, V. V. Konotop, B. Malomed, and V. M. Perez-Garcia, “A quasi-local Gross-Pitaevskii equation for attractive Bose-Einstein condensates,” Math. And Comp. in Sim. 6221–30 (2003).
[Crossref]

Nat. Photonics (1)

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

Nat. Phys. (1)

W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46–51 (2007).
[Crossref]

Nonlinearity (1)

O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Subcritical localization in the discrete nonlinear Schrödinger-equation with arbitrary power nonlinearity,” Nonlinearity 7, 205–218 (1994).
[Crossref]

Opt. Lett. (5)

Phys. Lett. A (2)

P. D. Miller and S. Kamvissis, “On the semiclassical limit of the focusing nonlinear Schrodinger equation,” Phys. Lett. A 247, 75–86 (1998).
[Crossref]

J. C. Bronski and J. N. Kutz, “Numerical simulation of the semi-classical limit of the focusing nonlinear Schrodinger equation,” Phys. Lett. A 254, 325–336 (1999).
[Crossref]

Phys. Rep. (1)

L. Berge, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303, 259–370, (1998).
[Crossref]

Phys. Rev. A (1)

G. A. El and A. L. Krylov, “General-solution of the Cauchy-problem for the defocusing NLS equation in the Whitham limit,” Phys. Rev. A 203, 77–82 (1995).

Phys. Rev. E (6)

P. D. Rasmussen, O. Bang, and W. Krolikowski, “Theory of nonlocal soliton interaction in nematic liquid crystals,” Phys. Rev. E 72066611 (2005).
[Crossref]

V. M. Perez-Garcia, V. V. Konotop, and J. J. Garcia-Ripoll, “Dynamics of quasicollapse in nonlinear Schrödinger equation with nonlocal interactions,” Phys. Rev. E 62, 4300–4308 (2000).
[Crossref]

O. Bang, W. Krolikowski, J. Wyller, and 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]

A. I. Yakimenko, Y. A. Zaliznyak, and Y. S. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E 71, 065603 (2005).
[Crossref]

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

Phys. Rev. Lett (1)

C. Rotschild, O. Cohen, O. Manela, and M. Segev, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett 95, 213904, (2005).
[Crossref] [PubMed]

Phys. Rev. Lett. (8)

K. D. Moll, A. L. Gaeta, and G. Fibich. “Self-similar optical wave collapse: observation of the Townes profile,” Phys. Rev. Lett. 90, 203902, (2003).
[Crossref] [PubMed]

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482, (1964).
[Crossref]

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

G. Fibich, “Small beam nonparaxiality arrests self-focusing of optical beams,” Phys. Rev. Lett. 76, 4356–4359, (1996).
[Crossref] [PubMed]

R. Camassa and D. D. Holm, “An integrable shallow water equation with peaked solitons,” Phys. Rev. Lett. 71, 1661–1664 (1993).
[Crossref] [PubMed]

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

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

J. E. Rothenberg and D. Grischkowsky, “Observation of the formation of an optical intensity shock and wave breaking in the nonlinear propagation of pulses in optical fibres,” Phys. Rev. Lett. 94, 040403 (2005).

Physica D (1)

M. Ablowitz, I. Bakirtaş, and B. Ilan, “Wave collapse in a class of nonlocal nonlinear Schrödinger equation,” Physica D 207, 230–253 (2005).
[Crossref]

PRL (1)

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

Science (1)

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

SIAM J. Appl. Math. (1)

G. Fibich and G. Papanicolaou, “Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension,” SIAM J. Appl. Math. 60, 183–240, (1999).
[Crossref]

Teor. Mat. Fiz. (1)

S. K. Turitsyn, “Spatial dispersion of nonlinearity and stability of multidimensional solitons,” Teor. Mat. Fiz. 64, 226–232, (1985).

Z. Phys. (1)

E. Madelung, “Quantetheorie in hydrodynamischer form’” Z. Phys. 40, 322–326 (1927).
[Crossref]

Zh. Eksp. Teor. Fiz. (1)

A. V. Gurevich and A. L. Krylov, “Nondissipative shock waves in media with positive dispersion,” Zh. Eksp. Teor. Fiz. 921684–1699 (1987).

Other (4)

A. Fratalocchi, C. Conti, G. Ruocco, and S. Trillo are preparing a manuscript to be called “Thermodynamics of soliton gases: phase transitions and shock waves”.

A. M. Kamchatnov, Nonlinear Periodic Waves and Their Modulations (World Scientific, Singapore2000).
[Crossref]

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo are preparing a manuscript to be called “Observation of a gradient catastrophe generating solitons”.

C. Sulem and P. L. Sulem., The nonlinear Schrodinger equation: self-focusing and wave collapse (Springer-Verlag, New York, 1999).

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Figures (8)

Fig. 1.
Fig. 1. Simulation results showing a Gaussian beam propagating in (a)local; (b)weakly nonlocal (w=a/5); and (c) highly nonlocal (w=5a) media. In the local case, the beam narrows and generates a train of solitons; in the highly nonlocal case, the NLS is reduced to a linear harmonic oscillator equation and the beam width and intensity bounce as a function of distance; the weakly nonlocal case is in between the two extremes.
Fig. 2.
Fig. 2. Numerical results of beam profiles and wavefront forces (a-b) before, (c-d) near and (e-f) after peakon profile in the case of local nonlinearity. Top row: intensity (normalized to peak intensity in each case). Bottom row: derivative of total pressure (quantum pressure and nonlinear index change in Eq. (4)); inset: derivative of nonlinear index change term only. (a) Before the peakon, the Gaussian beam is wider in the center and lower in the tails and (b) focusing nonlinear index change dominates the force and compresses the beam inwards. (c) As the beam propagates, the beam matches the peakon profile exactly while (d) the force of the quantum pressure term becomes comparable to that of the nonlinear index change in the center. (e) Past the peakon, the profile continues to steepen and becomes narrower in the center and higher in the tails while (f) the derivative of the quantum pressure term overtakes the focusing index change near the center, creating a defocusing force near the origin while the rest of the beam continues to focus inwards.
Fig. 3.
Fig. 3. Simulation results showing peakon profile with increasing nonlocality (a,c,e) in the presence of noise(b,d,f). In the local case (a), the peakon profile is obtained, but is unstable to noise(b); Using a weak nonlocality(c) of w=a/3, the profile is also obtained, albeit with a slightly rounded peak, and is stable to noise(d); with high nonlocality (e-f) of w=2a, the peakon profile is totally lost. Figures are normalized to the maximum intensity in each case.
Fig. 4.
Fig. 4. Simulation results using different nonlocal kernels — (a) Gaussian, (b) Lorentzian, (c) exp(-|x|), and (d) solving the 2D heat diffusion equation (6) explicitly. Peakon-like profiles are obtained in each case at the same propagation distance, albeit with different input powers required.
Fig. 5.
Fig. 5. Results of 2+1D simulation of NLS with nonlocal nonlinearity (w=a/5) showing collapse-bounce cycles of an initial Gaussian beam. The beam starts off as a Gaussian (a). As it self-focuses, it narrows (b) and approaches the peakon profile (c), after which it continues to focus into a pedestal-shape profile (d). It then defocuses into a ring-shaped profile (e) before focusing again (f) in an oscillatory manner. The focusing-defocusing bounce cycles continue quasi-periodically throughout the beam propagation.
Fig. 6.
Fig. 6. Experimental results of self-focusing of a 2D Gaussian beam in a nonlocal nonlinear medium. Shown are experimental output pictures at (a) 200mW, (b) 400mW, (c) 510mW, and (d) 600mW. (e) Cross-sectional profiles of (a-d), individually normalized to peak power.
Fig. 7.
Fig. 7. Transverse profile of experimentally-observed peakon in Fig. 6(c) showing best fits to hyperbolic secant, Townes and a peakon exp(-|x|) profiles.
Fig. 8.
Fig. 8. Experimental output pictures for an initial Gaussian beam with power (a) 600mW, (b) 800mW, (c) 1300mW, (d) 1500mW, (e) 1700mW, and (f) 1900mW. The beam focuses, defocuses, and focuses in an oscillatory, quasi-periodic fashion.

Equations (6)

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i ψ z + 1 2 k 0 2 ψ + k 0 Δ n ( ψ 2 ) n 0 ψ = 0
ρ z + ( ρ v ) = 0
S z + 1 2 v 2 = Δ n ( ρ ) n 0 + ( 1 2 ρ 2 ρ )
v z + v v = [ Δ n ( ρ ) n 0 ] + ( 1 2 ρ 2 ρ )
κ 2 Δ n = α β ψ 2
2 ( Δ n ( x , y ) ) C Δ n ( x , y ) = κ ̅ I ( x , y )

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