Abstract

We discuss generic properties of rotating nonlinear wave solutions, the so called azimuthons, in nonlocal media. Variational methods allow us to derive approximative values for the rotating frequency, which is shown to depend crucially on the nonlocal response function. Further on, we link families of azimuthons to internal modes of classical non-rotating stationary solutions, namely vortex and multipole solitons. This offers an exhaustive method to identify azimuthons in a given nonlocal medium.

© 2008 Optical Society of America

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  1. G. I. Stegeman and M. Segev, "Optical spatial solitons and their interactions: universality and diversity," Science 286, 1518-1523 (1999).
    [CrossRef]
  2. S. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003).
  3. L. Berge, "Wave collapse in physics: principles and applications to light and plasma waves - weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation," Phys. Rep. 303, 259-370 (1998).
  4. Y. S. Kivshar and D. E. Pelinovsky, "Self-focusing and transverse instabilities of solitary waves," Phys. Rep. 331, 118-195 (2000).
    [CrossRef]
  5. V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, "The theory of spitral laser beams in nonlinear media," J. Mod. Opt. 39, 2277-2291 (1992).
    [CrossRef]
  6. D. Skryabin and W. Firth, "Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media," Phys. Rev. E 58, 3916-3930 (1998).
    [CrossRef]
  7. D. Suter and T. Blasberg, "Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium," Phys. Rev. A 48, 4583-4587 (1993).
    [CrossRef] [PubMed]
  8. S. Skupin,W. Krolikowski, and M. Saffman, "Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor," Phys. Rev. Lett. 98, 263902 (2007).
    [CrossRef] [PubMed]
  9. A. G. Litvak, V. Mironov, G. Fraiman, and A. Yunakovskii, "Thermal self-effect of wave beams in plasma with a nonlocal nonlinearity," Sov. J. Plasma Phys. 1, 31-37 (1975).
  10. C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, "Long-range interactions between optical solitons," Nature Phys. 2, 769-774 (2006).
    [CrossRef]
  11. G. Assanto and M. Peccianti, "Spatial solitons in nematic liquid crystals," IEEE J. Quantum Electron. 39, 13-21 (2003).
    [CrossRef]
  12. C. Conti, M. Peccianti, and G. Assanto, "Observation of optical spatial solitons in a highly nonlocal medium," Phys. Rev. Lett. 92, 113902 (2004).
    [CrossRef] [PubMed]
  13. M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, "Tunable refraction and reflection of self-confined light beams," Nature Phys. 2, 737-742 (2006).
    [CrossRef]
  14. J. Denschlag, J. Simsarian, D. Feder, C. Clark, L. Collins, J. Cubizolles, L. Deng, E. Hagley, K. Helmerson, W. Reinhardt, S. Rolston, B. Schneider, and W. Phillips, "Generating solitons by phase engineering of a Bose-Einstein condensate," Science 287, 97-101 (2000).
    [CrossRef]
  15. P. Pedri and L. Santos, "Two-dimensional bright solitons in dipolar Bose-Einstein condensates," Phys. Rev. Lett. 95, 200404 (2005).
    [CrossRef] [PubMed]
  16. T. Koch, T. Lahaye, J. Metz, A. Frohlich, B. Griesmaier, and T. Pfau, "Stabilization of a purely dipolar quantum gas against collapse," Nature Phys. 4, 218-222 (2008).
    [CrossRef]
  17. W. Kr???olikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
    [CrossRef]
  18. S. Turitsyn, "Spatial dispersion of nonlinearity and stability of multidimensional solitons," Theor. Math. Phys. 64, 797-801 (1985).
    [CrossRef]
  19. 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]
  20. D. Briedis, D. Edmundson, O. Bang, and W. Krolikowski, "Ring vortex solitons in nonlocal nonlinear media," Opt. Express 13, 435-443 (2005).
    [CrossRef] [PubMed]
  21. 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]
  22. 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]
  23. A. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, "Dynamics of two-dimensional coherent structures in nonlocal nonlinear media," Phys. Rev. E 73, 066605 (2006).
    [CrossRef]
  24. Y. V. Kartashov, L. Torner, V. A. Vysloukh, and D. Mihalache, "Multipole vector solitons in nonlocal nonlinear media," Opt. Lett. 31, 1483-1485 (2006).
    [CrossRef] [PubMed]
  25. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Stability of vortex solitons in thermal nonlinear media," Opt. Express 15, 9378-9384 (2007).
    [CrossRef] [PubMed]
  26. Yu. A. Zaliznyak, A. I. Yakimenko, "Three-dimensional solitons and vortices in dipolar Bose-Einstein condensates," Phys. Lett. A 372, 2862-2866 (2008).
    [CrossRef]
  27. S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, "Stable rotating dipole solitons in nonlocal optical media," Opt. Lett. 31, 1100-1102 (2006).
    [CrossRef] [PubMed]
  28. A. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, "Azimuthons: spatially modulated vortex solitons," Phys. Rev. Lett. 95, 203904 (2005).
    [CrossRef] [PubMed]
  29. P. Coullet, L. Gil, and F. Rocca, "Optical vortices," Opt. Commun. 73, 403-408 (1989).
    [CrossRef]
  30. A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, "Optical vortices and vortex solitons," Prog. Optics 47, 291-391 (2005).
    [CrossRef]
  31. D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, "Spiraling multivortex solitons in nonlocal nonlinear media," Opt. Lett. 33, 198-200 (2008).
    [CrossRef] [PubMed]
  32. S. Lopez-Aguayo, A. S. Desyatnikov, and Y. S. Kivshar, "Azimuthons in nonlocal nonlinear media," Opt. Express 14, 7903-7908 (2006).
    [CrossRef] [PubMed]
  33. D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, "Laguerre and Hermite soliton clusters in nonlocal nonlinear media," Phys. Rev. Lett. 98, 053901 (2007).
    [CrossRef] [PubMed]
  34. E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A. Pure. Appl. Opt. 6, S157- S161 (2004).
    [CrossRef]
  35. D. Skryabin, J. McSloy, and W. Firth, "Stability of spiralling solitary waves in Hamiltonian systems," Phys. Rev. E 66, 055602 (2002).
    [CrossRef]
  36. N. N. Rozanov, "On the Translational and Rotational Motion of Nonlinear Optical Structures as a Whole," Opt. Spectrosc. 96, 405-408 (2004).
    [CrossRef]
  37. D. Anderson, "Variational approach to nonlinear pulse propagation in optical fibers," Phys. Rev. A 27, 3135-3145 (1983).
    [CrossRef]
  38. D. Skryabin and A. Vladimirov, "Vortex Induced Rotation of clusters of localized states in the complex Ginzburg-Landau equation," Phys. Rev. Lett. 89, 044101 (2002).
    [CrossRef] [PubMed]
  39. N. Rosanov, S. Fedorov, and A. Shatsev, "Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling," Phys. Rev. Lett. 95, 053903 (2005).
    [CrossRef] [PubMed]

2008 (3)

T. Koch, T. Lahaye, J. Metz, A. Frohlich, B. Griesmaier, and T. Pfau, "Stabilization of a purely dipolar quantum gas against collapse," Nature Phys. 4, 218-222 (2008).
[CrossRef]

Yu. A. Zaliznyak, A. I. Yakimenko, "Three-dimensional solitons and vortices in dipolar Bose-Einstein condensates," Phys. Lett. A 372, 2862-2866 (2008).
[CrossRef]

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, "Spiraling multivortex solitons in nonlocal nonlinear media," Opt. Lett. 33, 198-200 (2008).
[CrossRef] [PubMed]

2007 (3)

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Stability of vortex solitons in thermal nonlinear media," Opt. Express 15, 9378-9384 (2007).
[CrossRef] [PubMed]

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, "Laguerre and Hermite soliton clusters in nonlocal nonlinear media," Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

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

2006 (7)

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, "Long-range interactions between optical solitons," Nature Phys. 2, 769-774 (2006).
[CrossRef]

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, "Tunable refraction and reflection of self-confined light beams," Nature Phys. 2, 737-742 (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. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, "Dynamics of two-dimensional coherent structures in nonlocal nonlinear media," Phys. Rev. E 73, 066605 (2006).
[CrossRef]

S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, "Stable rotating dipole solitons in nonlocal optical media," Opt. Lett. 31, 1100-1102 (2006).
[CrossRef] [PubMed]

Y. V. Kartashov, L. Torner, V. A. Vysloukh, and D. Mihalache, "Multipole vector solitons in nonlocal nonlinear media," Opt. Lett. 31, 1483-1485 (2006).
[CrossRef] [PubMed]

S. Lopez-Aguayo, A. S. Desyatnikov, and Y. S. Kivshar, "Azimuthons in nonlocal nonlinear media," Opt. Express 14, 7903-7908 (2006).
[CrossRef] [PubMed]

2005 (6)

D. Briedis, D. Edmundson, O. Bang, and W. Krolikowski, "Ring vortex solitons in nonlocal nonlinear media," Opt. Express 13, 435-443 (2005).
[CrossRef] [PubMed]

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, "Optical vortices and vortex solitons," Prog. Optics 47, 291-391 (2005).
[CrossRef]

A. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, "Azimuthons: spatially modulated vortex solitons," Phys. Rev. Lett. 95, 203904 (2005).
[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]

P. Pedri and L. Santos, "Two-dimensional bright solitons in dipolar Bose-Einstein condensates," Phys. Rev. Lett. 95, 200404 (2005).
[CrossRef] [PubMed]

N. Rosanov, S. Fedorov, and A. Shatsev, "Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling," Phys. Rev. Lett. 95, 053903 (2005).
[CrossRef] [PubMed]

2004 (3)

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

E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A. Pure. Appl. Opt. 6, S157- S161 (2004).
[CrossRef]

N. N. Rozanov, "On the Translational and Rotational Motion of Nonlinear Optical Structures as a Whole," Opt. Spectrosc. 96, 405-408 (2004).
[CrossRef]

2003 (1)

G. Assanto and M. Peccianti, "Spatial solitons in nematic liquid crystals," IEEE J. Quantum Electron. 39, 13-21 (2003).
[CrossRef]

2002 (3)

D. Skryabin and A. Vladimirov, "Vortex Induced Rotation of clusters of localized states in the complex Ginzburg-Landau equation," Phys. Rev. Lett. 89, 044101 (2002).
[CrossRef] [PubMed]

D. Skryabin, J. McSloy, and W. Firth, "Stability of spiralling solitary waves in Hamiltonian systems," Phys. Rev. E 66, 055602 (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 (1)

W. Kr???olikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
[CrossRef]

2000 (2)

J. Denschlag, J. Simsarian, D. Feder, C. Clark, L. Collins, J. Cubizolles, L. Deng, E. Hagley, K. Helmerson, W. Reinhardt, S. Rolston, B. Schneider, and W. Phillips, "Generating solitons by phase engineering of a Bose-Einstein condensate," Science 287, 97-101 (2000).
[CrossRef]

Y. S. Kivshar and D. E. Pelinovsky, "Self-focusing and transverse instabilities of solitary waves," Phys. Rep. 331, 118-195 (2000).
[CrossRef]

1999 (1)

G. I. Stegeman and M. Segev, "Optical spatial solitons and their interactions: universality and diversity," Science 286, 1518-1523 (1999).
[CrossRef]

1998 (2)

L. Berge, "Wave collapse in physics: principles and applications to light and plasma waves - weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation," Phys. Rep. 303, 259-370 (1998).

D. Skryabin and W. Firth, "Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media," Phys. Rev. E 58, 3916-3930 (1998).
[CrossRef]

1993 (1)

D. Suter and T. Blasberg, "Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium," Phys. Rev. A 48, 4583-4587 (1993).
[CrossRef] [PubMed]

1992 (1)

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, "The theory of spitral laser beams in nonlinear media," J. Mod. Opt. 39, 2277-2291 (1992).
[CrossRef]

1989 (1)

P. Coullet, L. Gil, and F. Rocca, "Optical vortices," Opt. Commun. 73, 403-408 (1989).
[CrossRef]

1985 (1)

S. Turitsyn, "Spatial dispersion of nonlinearity and stability of multidimensional solitons," Theor. Math. Phys. 64, 797-801 (1985).
[CrossRef]

1983 (1)

D. Anderson, "Variational approach to nonlinear pulse propagation in optical fibers," Phys. Rev. A 27, 3135-3145 (1983).
[CrossRef]

1975 (1)

A. G. Litvak, V. Mironov, G. Fraiman, and A. Yunakovskii, "Thermal self-effect of wave beams in plasma with a nonlocal nonlinearity," Sov. J. Plasma Phys. 1, 31-37 (1975).

Abramochkin, E. G.

E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A. Pure. Appl. Opt. 6, S157- S161 (2004).
[CrossRef]

Alfassi, B.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, "Long-range interactions between optical solitons," Nature Phys. 2, 769-774 (2006).
[CrossRef]

Anderson, D.

D. Anderson, "Variational approach to nonlinear pulse propagation in optical fibers," Phys. Rev. A 27, 3135-3145 (1983).
[CrossRef]

Assanto, G.

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, "Tunable refraction and reflection of self-confined light beams," Nature Phys. 2, 737-742 (2006).
[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] [PubMed]

G. Assanto and M. Peccianti, "Spatial solitons in nematic liquid crystals," IEEE J. Quantum Electron. 39, 13-21 (2003).
[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]

S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, "Stable rotating dipole solitons in nonlocal optical media," Opt. Lett. 31, 1100-1102 (2006).
[CrossRef] [PubMed]

D. Briedis, D. Edmundson, O. Bang, and W. Krolikowski, "Ring vortex solitons in nonlocal nonlinear media," Opt. Express 13, 435-443 (2005).
[CrossRef] [PubMed]

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]

W. Kr???olikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
[CrossRef]

Berge, L.

L. Berge, "Wave collapse in physics: principles and applications to light and plasma waves - weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation," Phys. Rep. 303, 259-370 (1998).

Blasberg, T.

D. Suter and T. Blasberg, "Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium," Phys. Rev. A 48, 4583-4587 (1993).
[CrossRef] [PubMed]

Briedis, D.

Buccoliero, D.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, "Spiraling multivortex solitons in nonlocal nonlinear media," Opt. Lett. 33, 198-200 (2008).
[CrossRef] [PubMed]

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, "Laguerre and Hermite soliton clusters in nonlocal nonlinear media," Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

Clark, C.

J. Denschlag, J. Simsarian, D. Feder, C. Clark, L. Collins, J. Cubizolles, L. Deng, E. Hagley, K. Helmerson, W. Reinhardt, S. Rolston, B. Schneider, and W. Phillips, "Generating solitons by phase engineering of a Bose-Einstein condensate," Science 287, 97-101 (2000).
[CrossRef]

Cohen, O.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, "Long-range interactions between optical solitons," Nature Phys. 2, 769-774 (2006).
[CrossRef]

Collins, L.

J. Denschlag, J. Simsarian, D. Feder, C. Clark, L. Collins, J. Cubizolles, L. Deng, E. Hagley, K. Helmerson, W. Reinhardt, S. Rolston, B. Schneider, and W. Phillips, "Generating solitons by phase engineering of a Bose-Einstein condensate," Science 287, 97-101 (2000).
[CrossRef]

Conti, C.

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

Coullet, P.

P. Coullet, L. Gil, and F. Rocca, "Optical vortices," Opt. Commun. 73, 403-408 (1989).
[CrossRef]

Cubizolles, J.

J. Denschlag, J. Simsarian, D. Feder, C. Clark, L. Collins, J. Cubizolles, L. Deng, E. Hagley, K. Helmerson, W. Reinhardt, S. Rolston, B. Schneider, and W. Phillips, "Generating solitons by phase engineering of a Bose-Einstein condensate," Science 287, 97-101 (2000).
[CrossRef]

Deng, L.

J. Denschlag, J. Simsarian, D. Feder, C. Clark, L. Collins, J. Cubizolles, L. Deng, E. Hagley, K. Helmerson, W. Reinhardt, S. Rolston, B. Schneider, and W. Phillips, "Generating solitons by phase engineering of a Bose-Einstein condensate," Science 287, 97-101 (2000).
[CrossRef]

Denschlag, J.

J. Denschlag, J. Simsarian, D. Feder, C. Clark, L. Collins, J. Cubizolles, L. Deng, E. Hagley, K. Helmerson, W. Reinhardt, S. Rolston, B. Schneider, and W. Phillips, "Generating solitons by phase engineering of a Bose-Einstein condensate," Science 287, 97-101 (2000).
[CrossRef]

Desyatnikov, A.

A. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, "Azimuthons: spatially modulated vortex solitons," Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef] [PubMed]

Desyatnikov, A. S.

Dyadyusha, A.

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, "Tunable refraction and reflection of self-confined light beams," Nature Phys. 2, 737-742 (2006).
[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]

D. Briedis, D. Edmundson, O. Bang, and W. Krolikowski, "Ring vortex solitons in nonlocal nonlinear media," Opt. Express 13, 435-443 (2005).
[CrossRef] [PubMed]

Feder, D.

J. Denschlag, J. Simsarian, D. Feder, C. Clark, L. Collins, J. Cubizolles, L. Deng, E. Hagley, K. Helmerson, W. Reinhardt, S. Rolston, B. Schneider, and W. Phillips, "Generating solitons by phase engineering of a Bose-Einstein condensate," Science 287, 97-101 (2000).
[CrossRef]

Fedorov, S.

N. Rosanov, S. Fedorov, and A. Shatsev, "Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling," Phys. Rev. Lett. 95, 053903 (2005).
[CrossRef] [PubMed]

Firth, W.

D. Skryabin, J. McSloy, and W. Firth, "Stability of spiralling solitary waves in Hamiltonian systems," Phys. Rev. E 66, 055602 (2002).
[CrossRef]

D. Skryabin and W. Firth, "Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media," Phys. Rev. E 58, 3916-3930 (1998).
[CrossRef]

Fraiman, G.

A. G. Litvak, V. Mironov, G. Fraiman, and A. Yunakovskii, "Thermal self-effect of wave beams in plasma with a nonlocal nonlinearity," Sov. J. Plasma Phys. 1, 31-37 (1975).

Frohlich, A.

T. Koch, T. Lahaye, J. Metz, A. Frohlich, B. Griesmaier, and T. Pfau, "Stabilization of a purely dipolar quantum gas against collapse," Nature Phys. 4, 218-222 (2008).
[CrossRef]

Gil, L.

P. Coullet, L. Gil, and F. Rocca, "Optical vortices," Opt. Commun. 73, 403-408 (1989).
[CrossRef]

Griesmaier, B.

T. Koch, T. Lahaye, J. Metz, A. Frohlich, B. Griesmaier, and T. Pfau, "Stabilization of a purely dipolar quantum gas against collapse," Nature Phys. 4, 218-222 (2008).
[CrossRef]

Hagley, E.

J. Denschlag, J. Simsarian, D. Feder, C. Clark, L. Collins, J. Cubizolles, L. Deng, E. Hagley, K. Helmerson, W. Reinhardt, S. Rolston, B. Schneider, and W. Phillips, "Generating solitons by phase engineering of a Bose-Einstein condensate," Science 287, 97-101 (2000).
[CrossRef]

Helmerson, K.

J. Denschlag, J. Simsarian, D. Feder, C. Clark, L. Collins, J. Cubizolles, L. Deng, E. Hagley, K. Helmerson, W. Reinhardt, S. Rolston, B. Schneider, and W. Phillips, "Generating solitons by phase engineering of a Bose-Einstein condensate," Science 287, 97-101 (2000).
[CrossRef]

Kaczmarek, M.

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, "Tunable refraction and reflection of self-confined light beams," Nature Phys. 2, 737-742 (2006).
[CrossRef]

Kartashov, Y. V.

Kivshar, Y. S.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, "Spiraling multivortex solitons in nonlocal nonlinear media," Opt. Lett. 33, 198-200 (2008).
[CrossRef] [PubMed]

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, "Laguerre and Hermite soliton clusters in nonlocal nonlinear media," Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, "Stable rotating dipole solitons in nonlocal optical media," Opt. Lett. 31, 1100-1102 (2006).
[CrossRef] [PubMed]

S. Lopez-Aguayo, A. S. Desyatnikov, and Y. S. Kivshar, "Azimuthons in nonlocal nonlinear media," Opt. Express 14, 7903-7908 (2006).
[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]

A. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, "Azimuthons: spatially modulated vortex solitons," Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef] [PubMed]

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, "Optical vortices and vortex solitons," Prog. Optics 47, 291-391 (2005).
[CrossRef]

Y. S. Kivshar and D. E. Pelinovsky, "Self-focusing and transverse instabilities of solitary waves," Phys. Rep. 331, 118-195 (2000).
[CrossRef]

Koch, T.

T. Koch, T. Lahaye, J. Metz, A. Frohlich, B. Griesmaier, and T. Pfau, "Stabilization of a purely dipolar quantum gas against collapse," Nature Phys. 4, 218-222 (2008).
[CrossRef]

Kr???olikowski, W.

W. Kr???olikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
[CrossRef]

Krolikowski, W.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, "Spiraling multivortex solitons in nonlocal nonlinear media," Opt. Lett. 33, 198-200 (2008).
[CrossRef] [PubMed]

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

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, "Laguerre and Hermite soliton clusters in nonlocal nonlinear media," Phys. Rev. Lett. 98, 053901 (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]

S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, "Stable rotating dipole solitons in nonlocal optical media," Opt. Lett. 31, 1100-1102 (2006).
[CrossRef] [PubMed]

D. Briedis, D. Edmundson, O. Bang, and W. Krolikowski, "Ring vortex solitons in nonlocal nonlinear media," Opt. Express 13, 435-443 (2005).
[CrossRef] [PubMed]

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]

Kruglov, V. I.

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, "The theory of spitral laser beams in nonlinear media," J. Mod. Opt. 39, 2277-2291 (1992).
[CrossRef]

Lahaye, T.

T. Koch, T. Lahaye, J. Metz, A. Frohlich, B. Griesmaier, and T. Pfau, "Stabilization of a purely dipolar quantum gas against collapse," Nature Phys. 4, 218-222 (2008).
[CrossRef]

Lashkin, V. M.

A. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, "Dynamics of two-dimensional coherent structures in nonlocal nonlinear media," Phys. Rev. E 73, 066605 (2006).
[CrossRef]

Litvak, A. G.

A. G. Litvak, V. Mironov, G. Fraiman, and A. Yunakovskii, "Thermal self-effect of wave beams in plasma with a nonlocal nonlinearity," Sov. J. Plasma Phys. 1, 31-37 (1975).

Logvin, Y. A.

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, "The theory of spitral laser beams in nonlinear media," J. Mod. Opt. 39, 2277-2291 (1992).
[CrossRef]

Lopez-Aguayo, S.

McSloy, J.

D. Skryabin, J. McSloy, and W. Firth, "Stability of spiralling solitary waves in Hamiltonian systems," Phys. Rev. E 66, 055602 (2002).
[CrossRef]

Metz, J.

T. Koch, T. Lahaye, J. Metz, A. Frohlich, B. Griesmaier, and T. Pfau, "Stabilization of a purely dipolar quantum gas against collapse," Nature Phys. 4, 218-222 (2008).
[CrossRef]

Mihalache, D.

Mironov, V.

A. G. Litvak, V. Mironov, G. Fraiman, and A. Yunakovskii, "Thermal self-effect of wave beams in plasma with a nonlocal nonlinearity," Sov. J. Plasma Phys. 1, 31-37 (1975).

Peccianti, M.

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, "Tunable refraction and reflection of self-confined light beams," Nature Phys. 2, 737-742 (2006).
[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] [PubMed]

G. Assanto and M. Peccianti, "Spatial solitons in nematic liquid crystals," IEEE J. Quantum Electron. 39, 13-21 (2003).
[CrossRef]

Pedri, P.

P. Pedri and L. Santos, "Two-dimensional bright solitons in dipolar Bose-Einstein condensates," Phys. Rev. Lett. 95, 200404 (2005).
[CrossRef] [PubMed]

Pelinovsky, D. E.

Y. S. Kivshar and D. E. Pelinovsky, "Self-focusing and transverse instabilities of solitary waves," Phys. Rep. 331, 118-195 (2000).
[CrossRef]

Pfau, T.

T. Koch, T. Lahaye, J. Metz, A. Frohlich, B. Griesmaier, and T. Pfau, "Stabilization of a purely dipolar quantum gas against collapse," Nature Phys. 4, 218-222 (2008).
[CrossRef]

Phillips, W.

J. Denschlag, J. Simsarian, D. Feder, C. Clark, L. Collins, J. Cubizolles, L. Deng, E. Hagley, K. Helmerson, W. Reinhardt, S. Rolston, B. Schneider, and W. Phillips, "Generating solitons by phase engineering of a Bose-Einstein condensate," Science 287, 97-101 (2000).
[CrossRef]

Prikhodko, O. O.

A. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, "Dynamics of two-dimensional coherent structures in nonlocal nonlinear media," Phys. Rev. E 73, 066605 (2006).
[CrossRef]

Rasmussen, J. J.

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]

W. Kr???olikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
[CrossRef]

Reinhardt, W.

J. Denschlag, J. Simsarian, D. Feder, C. Clark, L. Collins, J. Cubizolles, L. Deng, E. Hagley, K. Helmerson, W. Reinhardt, S. Rolston, B. Schneider, and W. Phillips, "Generating solitons by phase engineering of a Bose-Einstein condensate," Science 287, 97-101 (2000).
[CrossRef]

Rocca, F.

P. Coullet, L. Gil, and F. Rocca, "Optical vortices," Opt. Commun. 73, 403-408 (1989).
[CrossRef]

Rolston, S.

J. Denschlag, J. Simsarian, D. Feder, C. Clark, L. Collins, J. Cubizolles, L. Deng, E. Hagley, K. Helmerson, W. Reinhardt, S. Rolston, B. Schneider, and W. Phillips, "Generating solitons by phase engineering of a Bose-Einstein condensate," Science 287, 97-101 (2000).
[CrossRef]

Rosanov, N.

N. Rosanov, S. Fedorov, and A. Shatsev, "Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling," Phys. Rev. Lett. 95, 053903 (2005).
[CrossRef] [PubMed]

Rotschild, C.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, "Long-range interactions between optical solitons," Nature Phys. 2, 769-774 (2006).
[CrossRef]

Rozanov, N. N.

N. N. Rozanov, "On the Translational and Rotational Motion of Nonlinear Optical Structures as a Whole," Opt. Spectrosc. 96, 405-408 (2004).
[CrossRef]

Saffman, M.

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

Santos, L.

P. Pedri and L. Santos, "Two-dimensional bright solitons in dipolar Bose-Einstein condensates," Phys. Rev. Lett. 95, 200404 (2005).
[CrossRef] [PubMed]

Schneider, B.

J. Denschlag, J. Simsarian, D. Feder, C. Clark, L. Collins, J. Cubizolles, L. Deng, E. Hagley, K. Helmerson, W. Reinhardt, S. Rolston, B. Schneider, and W. Phillips, "Generating solitons by phase engineering of a Bose-Einstein condensate," Science 287, 97-101 (2000).
[CrossRef]

Segev, M.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, "Long-range interactions between optical solitons," Nature Phys. 2, 769-774 (2006).
[CrossRef]

G. I. Stegeman and M. Segev, "Optical spatial solitons and their interactions: universality and diversity," Science 286, 1518-1523 (1999).
[CrossRef]

Shatsev, A.

N. Rosanov, S. Fedorov, and A. Shatsev, "Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling," Phys. Rev. Lett. 95, 053903 (2005).
[CrossRef] [PubMed]

Simsarian, J.

J. Denschlag, J. Simsarian, D. Feder, C. Clark, L. Collins, J. Cubizolles, L. Deng, E. Hagley, K. Helmerson, W. Reinhardt, S. Rolston, B. Schneider, and W. Phillips, "Generating solitons by phase engineering of a Bose-Einstein condensate," Science 287, 97-101 (2000).
[CrossRef]

Skryabin, D.

D. Skryabin, J. McSloy, and W. Firth, "Stability of spiralling solitary waves in Hamiltonian systems," Phys. Rev. E 66, 055602 (2002).
[CrossRef]

D. Skryabin and A. Vladimirov, "Vortex Induced Rotation of clusters of localized states in the complex Ginzburg-Landau equation," Phys. Rev. Lett. 89, 044101 (2002).
[CrossRef] [PubMed]

D. Skryabin and W. Firth, "Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media," Phys. Rev. E 58, 3916-3930 (1998).
[CrossRef]

Skupin, S.

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

S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, "Stable rotating dipole solitons in nonlocal optical media," Opt. Lett. 31, 1100-1102 (2006).
[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]

Stegeman, G. I.

G. I. Stegeman and M. Segev, "Optical spatial solitons and their interactions: universality and diversity," Science 286, 1518-1523 (1999).
[CrossRef]

Sukhorukov, A. A.

A. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, "Azimuthons: spatially modulated vortex solitons," Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef] [PubMed]

Suter, D.

D. Suter and T. Blasberg, "Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium," Phys. Rev. A 48, 4583-4587 (1993).
[CrossRef] [PubMed]

Torner, L.

Turitsyn, S.

S. Turitsyn, "Spatial dispersion of nonlinearity and stability of multidimensional solitons," Theor. Math. Phys. 64, 797-801 (1985).
[CrossRef]

Vladimirov, A.

D. Skryabin and A. Vladimirov, "Vortex Induced Rotation of clusters of localized states in the complex Ginzburg-Landau equation," Phys. Rev. Lett. 89, 044101 (2002).
[CrossRef] [PubMed]

Volkov, V. M.

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, "The theory of spitral laser beams in nonlinear media," J. Mod. Opt. 39, 2277-2291 (1992).
[CrossRef]

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A. Pure. Appl. Opt. 6, S157- S161 (2004).
[CrossRef]

Vysloukh, V. A.

Wyller, J.

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]

W. Kr???olikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
[CrossRef]

Yakimenko, A. I.

Yu. A. Zaliznyak, A. I. Yakimenko, "Three-dimensional solitons and vortices in dipolar Bose-Einstein condensates," Phys. Lett. A 372, 2862-2866 (2008).
[CrossRef]

A. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, "Dynamics of two-dimensional coherent structures in nonlocal nonlinear media," Phys. Rev. E 73, 066605 (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]

Yunakovskii, A.

A. G. Litvak, V. Mironov, G. Fraiman, and A. Yunakovskii, "Thermal self-effect of wave beams in plasma with a nonlocal nonlinearity," Sov. J. Plasma Phys. 1, 31-37 (1975).

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]

Zaliznyak, Yu. A.

Yu. A. Zaliznyak, A. I. Yakimenko, "Three-dimensional solitons and vortices in dipolar Bose-Einstein condensates," Phys. Lett. A 372, 2862-2866 (2008).
[CrossRef]

IEEE J. Quantum Electron. (1)

G. Assanto and M. Peccianti, "Spatial solitons in nematic liquid crystals," IEEE J. Quantum Electron. 39, 13-21 (2003).
[CrossRef]

J. Mod. Opt. (1)

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, "The theory of spitral laser beams in nonlinear media," J. Mod. Opt. 39, 2277-2291 (1992).
[CrossRef]

J. Opt. A. Pure. Appl. Opt. (1)

E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A. Pure. Appl. Opt. 6, S157- S161 (2004).
[CrossRef]

Nature Phys. (3)

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, "Tunable refraction and reflection of self-confined light beams," Nature Phys. 2, 737-742 (2006).
[CrossRef]

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, "Long-range interactions between optical solitons," Nature Phys. 2, 769-774 (2006).
[CrossRef]

T. Koch, T. Lahaye, J. Metz, A. Frohlich, B. Griesmaier, and T. Pfau, "Stabilization of a purely dipolar quantum gas against collapse," Nature Phys. 4, 218-222 (2008).
[CrossRef]

Opt. Commun. (1)

P. Coullet, L. Gil, and F. Rocca, "Optical vortices," Opt. Commun. 73, 403-408 (1989).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Opt. Spectrosc. (1)

N. N. Rozanov, "On the Translational and Rotational Motion of Nonlinear Optical Structures as a Whole," Opt. Spectrosc. 96, 405-408 (2004).
[CrossRef]

Phys. Lett. A (1)

Yu. A. Zaliznyak, A. I. Yakimenko, "Three-dimensional solitons and vortices in dipolar Bose-Einstein condensates," Phys. Lett. A 372, 2862-2866 (2008).
[CrossRef]

Phys. Rep. (2)

L. Berge, "Wave collapse in physics: principles and applications to light and plasma waves - weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation," Phys. Rep. 303, 259-370 (1998).

Y. S. Kivshar and D. E. Pelinovsky, "Self-focusing and transverse instabilities of solitary waves," Phys. Rep. 331, 118-195 (2000).
[CrossRef]

Phys. Rev. A (2)

D. Suter and T. Blasberg, "Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium," Phys. Rev. A 48, 4583-4587 (1993).
[CrossRef] [PubMed]

D. Anderson, "Variational approach to nonlinear pulse propagation in optical fibers," Phys. Rev. A 27, 3135-3145 (1983).
[CrossRef]

Phys. Rev. E (7)

D. Skryabin, J. McSloy, and W. Firth, "Stability of spiralling solitary waves in Hamiltonian systems," Phys. Rev. E 66, 055602 (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]

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]

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, V. M. Lashkin, and O. O. Prikhodko, "Dynamics of two-dimensional coherent structures in nonlocal nonlinear media," Phys. Rev. E 73, 066605 (2006).
[CrossRef]

D. Skryabin and W. Firth, "Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media," Phys. Rev. E 58, 3916-3930 (1998).
[CrossRef]

W. Kr???olikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
[CrossRef]

Phys. Rev. Lett. (7)

P. Pedri and L. Santos, "Two-dimensional bright solitons in dipolar Bose-Einstein condensates," Phys. Rev. Lett. 95, 200404 (2005).
[CrossRef] [PubMed]

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

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

A. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, "Azimuthons: spatially modulated vortex solitons," Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef] [PubMed]

D. Skryabin and A. Vladimirov, "Vortex Induced Rotation of clusters of localized states in the complex Ginzburg-Landau equation," Phys. Rev. Lett. 89, 044101 (2002).
[CrossRef] [PubMed]

N. Rosanov, S. Fedorov, and A. Shatsev, "Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling," Phys. Rev. Lett. 95, 053903 (2005).
[CrossRef] [PubMed]

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, "Laguerre and Hermite soliton clusters in nonlocal nonlinear media," Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

Prog. Optics (1)

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, "Optical vortices and vortex solitons," Prog. Optics 47, 291-391 (2005).
[CrossRef]

Science (2)

G. I. Stegeman and M. Segev, "Optical spatial solitons and their interactions: universality and diversity," Science 286, 1518-1523 (1999).
[CrossRef]

J. Denschlag, J. Simsarian, D. Feder, C. Clark, L. Collins, J. Cubizolles, L. Deng, E. Hagley, K. Helmerson, W. Reinhardt, S. Rolston, B. Schneider, and W. Phillips, "Generating solitons by phase engineering of a Bose-Einstein condensate," Science 287, 97-101 (2000).
[CrossRef]

Sov. J. Plasma Phys. (1)

A. G. Litvak, V. Mironov, G. Fraiman, and A. Yunakovskii, "Thermal self-effect of wave beams in plasma with a nonlocal nonlinearity," Sov. J. Plasma Phys. 1, 31-37 (1975).

Theor. Math. Phys. (1)

S. Turitsyn, "Spatial dispersion of nonlinearity and stability of multidimensional solitons," Theor. Math. Phys. 64, 797-801 (1985).
[CrossRef]

Other (1)

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

Supplementary Material (6)

» Media 1: AVI (14527 KB)     
» Media 2: AVI (3680 KB)     
» Media 3: AVI (14996 KB)     
» Media 4: AVI (3808 KB)     
» Media 5: AVI (14656 KB)     
» Media 6: AVI (3705 KB)     

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Figures (9)

Fig. 1.
Fig. 1.

Nonlocal model and variational approach: Solid lines show (a) width σ, (b) amplitude A, and (c) frequency Ω| α=1 for the Bessel response as functions of the propagation constant λ 0. Dashed lines represent the same quantities for the Gaussian nonlocal response. The inset in (c) shows Ω| α=1 for the latter response on more appropriate scales. While both, width σ and amplitude A, exhibit similar behavior for the two nonlocal models the rotation frequency Ω| α=1 shows a completely different dependence on λ 0 and drastic difference in magnitude for Gaussian and Bessel responses.

Fig. 2.
Fig. 2.

(a) Mass M and (b) quantity N cc - N cs for Bessel (solid lines) and Gaussian (dashed lines) nonlocal responses as functions of the propagation constant λ 0. It is obvious that the difference in the rotating frequency Ω| α=1 shown in Fig. 1(c) is due to N cc -N cs [see Eq. (11)]. (c) Spatial Fourier Transform (FT) of the Bessel (solid lines) and Gaussian (dashed lines) nonlocal response R as a function of the transverse wave-vector k = k x 2 + k y 2 . The degree of spatial averaging is obviously larger for the Gaussian than for the Lorentzian.

Fig. 3.
Fig. 3.

Gaussian nonlocal model, M=200, λ 0=11.5: Solid line shows the result of Eq. (11) computed from the variational vortex approximation. Crosses depict results obtained from direct numerical solutions. Dashed lines represent results of the linear perturbation analysis of dipole and vortex soliton, respectively (see Sec. 5).

Fig. 4.
Fig. 4.

Bessel nonlocal model, M=2000, λ 0=418: Same coding as in Fig. 3.

Fig. 5.
Fig. 5.

Dipole D and deformation d for the Gaussian nonlocal model, M=200, λ D =11.8. The deformation d has a dipole shape, rotated by π/2 with respect to D. We can construct an approximate azimuthon U=D+iΩd for small rotation frequencies Ω. The amplitude ratio max d/max D gives the slope of Ω(α) at α=0.

Fig. 6.
Fig. 6.

Same as Fig. 5 but for the Bessel nonlocal model, M=2000, λD =444.

Fig. 7.
Fig. 7.

Gaussian nonlocal model, M=200, λ 0=11.5: Solid line shows the component δV 1, dashed line the component δV 2, for m=2 and eigenvalue κ=-1.4 computed from Eq. (20). The movie (4 MB or 15 MB) shows the evolution of intensity and phase of the vortex perturbed with the eigenvector shown above, U| z=0=V exp()+δV 1 exp(i3ϕ)+δV 2 exp(-). The chosen amplitudes of the perturbation are such that α=0.85, the rotating frequency is close to Ω| α=1=0.7. [Media 1][Media 2]

Fig. 8.
Fig. 8.

Bessel nonlocal model, M=2000, λ 0=418: Same as in Fig. 7, m=2, κ=-62. The movie (4 MB or 15 MB) shows the evolution of the perturbed vortex with α=0.9, the rotating frequency is close to Ω| α=1=31. [Media 3][Media 4]

Fig. 9.
Fig. 9.

Gaussian nonlocal model, M=200, λ 0=11.5: Same as in Fig. 7, m=3, κ=3.8. The movie (4 MB or 15 MB) shows the evolution of perturbed vortex U| z=0=V exp()+δV 1 exp(i4ϕ)+δV 2 exp(-i2ϕ), α=0.9, the rotating frequency is close to Ω| α=1=-1.27. [Media 5][Media 6]

Equations (33)

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i z ψ + Δ ψ + θ ψ = 0 .
θ = 1 2 π e r r 2 2 ψ ( r , z ) 2 d 2 r ,
θ Δ θ = ψ 2 ,
θ = 1 2 π 𝔎 0 ( r r ) ψ ( r , z ) 2 d 2 r ,
θ = R ( r r ) ψ ( r , z ) 2 d 2 r ,
ψ ( r , ϕ , z ) = U ( r , ϕ Ω z ) e i λ z ,
λ M + Ω L z + I + N = 0
λ L z + Ω M + I + N = 0 .
M = U ( r ) 2 d 2 r
L z = i U * ( r ) ϕ U ( r ) d 2 r
I = U * ( r ) Δ U ( r ) d 2 r
N = R ( r r ) U ( r ) 2 U ( r ) 2 d 2 r d 2 r
M = ϕ U ( r ) 2 d 2 r
I = i [ ϕ U * ( r ) ] Δ U ( r ) d 2 r
N = i R ( r r ) U ( r ) 2 [ ϕ U * ( r ) ] U ( r ) d 2 r d 2 r .
Ω = M ( I + N ) L z ( I + N ) L z 2 M M .
U ( r , ϕ Ω z ) = AV ( r ) [ cos ( ϕ Ω z ) + i α sin ( ϕ Ω z ) ] ,
N cc = R ( r r ) V 2 ( r ) cos 2 ( ϕ ) V 2 ( r ) cos 2 ( ϕ ) d 2 r d 2 r
N cs = R ( r r ) V 2 ( r ) cos 2 ( ϕ ) V 2 ( r ) sin 2 ( ϕ ) d 2 r d 2 r .
Ω = 2 α α 2 + 1 2 ( N cc N cs ) M 0 .
λ 0 σ 8 + ( 2 + 2 λ 0 ) σ 6 + ( 8 + 4 λ 0 ) σ 4 8 σ 2 8 = 0 ,
A = 2 ( σ 2 + 1 ) σ 3 ( σ 2 + 1 ) ( λ 0 σ 2 + 2 ) σ 4 + 2 σ 2 + 2 ,
Ω = 2 α α 2 + 1 A 2 σ 8 8 ( σ 2 + 1 ) 3 .
U ( r , ϕ Ω z ) = D ( r , ϕ Ω z ) + i Ω d ( r , ϕ Ω z ) ,
λ D d + Δ d + R ( r r ) D 2 ( r ) d 2 r d = ϕ D .
ψ = ( V + δ V ) e i ϕ + i λ 0 z
[ i z λ 0 + 1 r r ( r r ) + 1 r 2 ( ϕ + i ) 2 + R ( r r ) V 2 ( r ) d 2 r ] δ V
+ V R ( r r ) V ( r ) [ δ V ( r , z ) + δ V * ( r , z ) ] d 2 r = 0 .
δ V = δ V 1 ( r ) e i m ϕ + i κ z + δ V 2 * ( r ) e i m ϕ i κ * z
L ̂ ( δ V 1 δ V 2 ) = κ ( δ V 1 δ V 2 ) ,
L ̂ ( V 1 V 2 ) = ( [ 1 r r ( r r ) ( m + 1 ) 2 r 2 λ 0 + R ( r r ) V 2 ( r ) d 2 r ] δ V 1 [ 1 r r ( r r ) ( m 1 ) 2 r 2 λ 0 + R ( r r ) V 2 ( r ) d 2 r ] δ V 2 )
+ ( V R ( r r ) V ( r ) [ δ V 1 ( r ) + δ V 2 ( r ) ] cos [ m ( ϕ ϕ ) ] d 2 r V R ( r r ) V ( r ) [ δ V 2 ( r ) + δ V 1 ( r ) ] cos [ m ( ϕ ϕ ) ] d 2 r ) .
U ( r , ϕ ) z = 0 = [ V ( r ) + δ V 1 ( r ) e im ϕ + δ V 2 ( r ) e im ϕ ] e i ϕ .

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