Abstract

We derive analytical solutions to the cubic-quintic nonlinear Schrödinger equation with potentials and nonlinearities depending on both propagation distance and transverse space. Among other, circle solitons and multi-peaked vortex solitons are found. These solitary waves propagate self-similarly and are characterized by three parameters, the modal numbers m and n, and the modulation depth of intensity. We find that the stable fundamental solitons with m = 0 and the low-order solitons with m = 1, ≤ 2 can be supported with the energy eigenvalues E = 0 and E ≠ 0. However, higher-order solitons display unstable propagation over prolonged distances. The stability of solutions is examined by numerical simulations.

© 2016 Optical Society of America

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References

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  1. N. J. Zabusky and M. D. Kruskal, “Interactions of solitons in a collisionless plasma and the recurrence of initial states,” Phys. Rev. Lett. 15(6), 240–243 (1965).
    [Crossref]
  2. W. J. Mullin and A. R. Sakhel, “Generalized Bose-Einstein condensation,” J. Low Temp. Phys. 166(3-4), 125–150 (2012).
    [Crossref]
  3. Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61(4), 763–915 (1989).
    [Crossref]
  4. C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, (Springer-Verlag, 2000)
  5. F. K. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length,” Phys. Rev. A 67(1), 013605 (2003).
    [Crossref]
  6. B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in periodic potentials,” Europhys. Lett. 63(5), 642–648 (2003).
    [Crossref]
  7. D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004).
    [Crossref] [PubMed]
  8. D. Mihalache, D. Mazilu, F. Lederer, and Y. S. Kivshar, “Spatiotemporal surface solitons in two-dimensional photonic lattices,” Opt. Lett. 32(21), 3173–3175 (2007).
    [Crossref] [PubMed]
  9. J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92(16), 163902 (2004).
    [Crossref] [PubMed]
  10. T. Mayteevarunyoo, B. A. Malomed, and A. Roeksabutr, “Solitons and vortices in nonlinear two-dimensional photonic crystals of the Kronig-Penney type,” Opt. Express 19(18), 17834–17851 (2011).
    [Crossref] [PubMed]
  11. S.-L. Xu and M. R. Belic, “Light bullets in three-dimensional complex Ginzburg-Landau equation with modulated Kummer-Gauss photonic lattice,” EPL 108(3), 34001 (2014).
    [Crossref]
  12. F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, “Light bullets in Bessel optical lattices with spatially modulated nonlinearity,” Opt. Express 17(14), 11328–11334 (2009).
    [Crossref] [PubMed]
  13. D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability limits for three-dimensional vortex solitons in the Ginzburg-Landau equation with the cubic-quintic nonlinearity,” Phys. Rev. A 76(4), 0453803 (2007).
    [Crossref]
  14. Z. Birnbaum and B. A. Malomed, “Families of spatial solitons in a two-channel waveguide with the cubic-quintic nonlinearity,” Physica D 237(24), 3252–3262 (2008).
    [Crossref]
  15. G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219(1-6), 427–433 (2003).
    [Crossref]
  16. V. N. Serkin, T. L. Belyaeva, I. V. Alexandrov, and G. M. Melchor, “Novel topological quasi-soliton solutions for the nonlinear cubic-quintic equation model,” Proc. SPIE 4271, 292–302 (2001).
    [Crossref]
  17. R. Hao, L. Li, Z. Li, R. Yang, and G. Zhou, “A new approach to exact soliton solutions and soliton interaction for the nonlinear Schrödinger equation with variable coefficients,” Opt. Commun. 245, 383 (2005).
    [Crossref]
  18. J.-R. He, L. Yi, and H.-M. Li, “Self-similar propagation and asymptotic optical waves in nonlinear waveguides,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(1), 013202 (2014).
    [Crossref] [PubMed]
  19. R. M. Caplan, R. Carretero-González, P. G. Kevrekidis, and B. A. Malomed, “Existence, stability, and scattering of bright vortices in the cubic–quintic nonlinear Schrödinger equation,” Math. Comput. Simul. 82(7), 1150–1171 (2012).
    [Crossref]
  20. S. L. Xu, G. P. Zhou, N. Z. Petrović, and M. R. Belić, “Two-dimensional dark solitons in diffusive nonlocal nonlinear media,” J. Opt. 17, 105605 (2015).
    [Crossref]
  21. S. L. Xu, J. C. Liang, and L. Yi, “Self-similar solitary waves in Bessel optical lattices,” J. Opt. Soc. Am. B 27(1), 99 (2010).
    [Crossref]
  22. J. Belmonte-Beitia and J. Cuevas, “Solitons for the cubic-quintic nonlinear Schrodinger equation with time- and space-modulated coefficients,” J. Phys. A 42, 11 (2009).
  23. M. V. Berry and M. R. Dennis, “Quantum cores of optical phase singularities,” J. Opt. A, Pure Appl. Opt. 6(5), S178–S180 (2004).
    [Crossref]
  24. J. C. Chen, X. F. Zhang, B. Li, and Y. Chen, “Exact solutions to the two-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrödinger equation with an external potential,” Chin. Phys. Lett. 29, 220 (2012).
  25. X. Y. Tang and P. K. Shukla, “Solution of the one-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrödinger equation with an external potential,” Phys. Rev. A 76(1), 013612 (2007).
    [Crossref]
  26. J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98(6), 064102 (2007).
    [Crossref] [PubMed]
  27. A. T. Avelar, D. Bazeia, and W. B. Cardoso, “Solitons with cubic and quintic nonlinearities modulated in space and time,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(2), 025602 (2009).
    [Crossref] [PubMed]
  28. V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous matter-wave solitons near the Feshbach resonance,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 81, 023610 (2010).
  29. D. Zwillinger, Handbook of Differential Equations, 3rd ed. (Academic Press, 1997).
  30. J. K. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118(2), 153 (2007).
    [Crossref]
  31. J. K. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM, 2010).
  32. A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. 95(20), 203904 (2005).
    [Crossref] [PubMed]
  33. M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation,” Phys. Rev. Lett. 101(12), 123904 (2008).
    [Crossref] [PubMed]

2015 (1)

S. L. Xu, G. P. Zhou, N. Z. Petrović, and M. R. Belić, “Two-dimensional dark solitons in diffusive nonlocal nonlinear media,” J. Opt. 17, 105605 (2015).
[Crossref]

2014 (2)

S.-L. Xu and M. R. Belic, “Light bullets in three-dimensional complex Ginzburg-Landau equation with modulated Kummer-Gauss photonic lattice,” EPL 108(3), 34001 (2014).
[Crossref]

J.-R. He, L. Yi, and H.-M. Li, “Self-similar propagation and asymptotic optical waves in nonlinear waveguides,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(1), 013202 (2014).
[Crossref] [PubMed]

2012 (3)

R. M. Caplan, R. Carretero-González, P. G. Kevrekidis, and B. A. Malomed, “Existence, stability, and scattering of bright vortices in the cubic–quintic nonlinear Schrödinger equation,” Math. Comput. Simul. 82(7), 1150–1171 (2012).
[Crossref]

W. J. Mullin and A. R. Sakhel, “Generalized Bose-Einstein condensation,” J. Low Temp. Phys. 166(3-4), 125–150 (2012).
[Crossref]

J. C. Chen, X. F. Zhang, B. Li, and Y. Chen, “Exact solutions to the two-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrödinger equation with an external potential,” Chin. Phys. Lett. 29, 220 (2012).

2011 (1)

2010 (2)

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous matter-wave solitons near the Feshbach resonance,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 81, 023610 (2010).

S. L. Xu, J. C. Liang, and L. Yi, “Self-similar solitary waves in Bessel optical lattices,” J. Opt. Soc. Am. B 27(1), 99 (2010).
[Crossref]

2009 (3)

J. Belmonte-Beitia and J. Cuevas, “Solitons for the cubic-quintic nonlinear Schrodinger equation with time- and space-modulated coefficients,” J. Phys. A 42, 11 (2009).

A. T. Avelar, D. Bazeia, and W. B. Cardoso, “Solitons with cubic and quintic nonlinearities modulated in space and time,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(2), 025602 (2009).
[Crossref] [PubMed]

F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, “Light bullets in Bessel optical lattices with spatially modulated nonlinearity,” Opt. Express 17(14), 11328–11334 (2009).
[Crossref] [PubMed]

2008 (2)

Z. Birnbaum and B. A. Malomed, “Families of spatial solitons in a two-channel waveguide with the cubic-quintic nonlinearity,” Physica D 237(24), 3252–3262 (2008).
[Crossref]

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation,” Phys. Rev. Lett. 101(12), 123904 (2008).
[Crossref] [PubMed]

2007 (5)

J. K. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118(2), 153 (2007).
[Crossref]

X. Y. Tang and P. K. Shukla, “Solution of the one-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrödinger equation with an external potential,” Phys. Rev. A 76(1), 013612 (2007).
[Crossref]

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98(6), 064102 (2007).
[Crossref] [PubMed]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability limits for three-dimensional vortex solitons in the Ginzburg-Landau equation with the cubic-quintic nonlinearity,” Phys. Rev. A 76(4), 0453803 (2007).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, and Y. S. Kivshar, “Spatiotemporal surface solitons in two-dimensional photonic lattices,” Opt. Lett. 32(21), 3173–3175 (2007).
[Crossref] [PubMed]

2005 (2)

R. Hao, L. Li, Z. Li, R. Yang, and G. Zhou, “A new approach to exact soliton solutions and soliton interaction for the nonlinear Schrödinger equation with variable coefficients,” Opt. Commun. 245, 383 (2005).
[Crossref]

A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. 95(20), 203904 (2005).
[Crossref] [PubMed]

2004 (3)

M. V. Berry and M. R. Dennis, “Quantum cores of optical phase singularities,” J. Opt. A, Pure Appl. Opt. 6(5), S178–S180 (2004).
[Crossref]

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92(16), 163902 (2004).
[Crossref] [PubMed]

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004).
[Crossref] [PubMed]

2003 (3)

F. K. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length,” Phys. Rev. A 67(1), 013605 (2003).
[Crossref]

B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in periodic potentials,” Europhys. Lett. 63(5), 642–648 (2003).
[Crossref]

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219(1-6), 427–433 (2003).
[Crossref]

2001 (1)

V. N. Serkin, T. L. Belyaeva, I. V. Alexandrov, and G. M. Melchor, “Novel topological quasi-soliton solutions for the nonlinear cubic-quintic equation model,” Proc. SPIE 4271, 292–302 (2001).
[Crossref]

1989 (1)

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61(4), 763–915 (1989).
[Crossref]

1965 (1)

N. J. Zabusky and M. D. Kruskal, “Interactions of solitons in a collisionless plasma and the recurrence of initial states,” Phys. Rev. Lett. 15(6), 240–243 (1965).
[Crossref]

Abdullaev, F. K.

F. K. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length,” Phys. Rev. A 67(1), 013605 (2003).
[Crossref]

Aitchison, J. S.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92(16), 163902 (2004).
[Crossref] [PubMed]

Alexander, T. J.

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004).
[Crossref] [PubMed]

Alexandrov, I. V.

V. N. Serkin, T. L. Belyaeva, I. V. Alexandrov, and G. M. Melchor, “Novel topological quasi-soliton solutions for the nonlinear cubic-quintic equation model,” Proc. SPIE 4271, 292–302 (2001).
[Crossref]

Avelar, A. T.

A. T. Avelar, D. Bazeia, and W. B. Cardoso, “Solitons with cubic and quintic nonlinearities modulated in space and time,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(2), 025602 (2009).
[Crossref] [PubMed]

Baizakov, B. B.

B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in periodic potentials,” Europhys. Lett. 63(5), 642–648 (2003).
[Crossref]

Bazeia, D.

A. T. Avelar, D. Bazeia, and W. B. Cardoso, “Solitons with cubic and quintic nonlinearities modulated in space and time,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(2), 025602 (2009).
[Crossref] [PubMed]

Belic, M.

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation,” Phys. Rev. Lett. 101(12), 123904 (2008).
[Crossref] [PubMed]

Belic, M. R.

S. L. Xu, G. P. Zhou, N. Z. Petrović, and M. R. Belić, “Two-dimensional dark solitons in diffusive nonlocal nonlinear media,” J. Opt. 17, 105605 (2015).
[Crossref]

S.-L. Xu and M. R. Belic, “Light bullets in three-dimensional complex Ginzburg-Landau equation with modulated Kummer-Gauss photonic lattice,” EPL 108(3), 34001 (2014).
[Crossref]

Belmonte-Beitia, J.

J. Belmonte-Beitia and J. Cuevas, “Solitons for the cubic-quintic nonlinear Schrodinger equation with time- and space-modulated coefficients,” J. Phys. A 42, 11 (2009).

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98(6), 064102 (2007).
[Crossref] [PubMed]

Belyaeva, T. L.

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous matter-wave solitons near the Feshbach resonance,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 81, 023610 (2010).

V. N. Serkin, T. L. Belyaeva, I. V. Alexandrov, and G. M. Melchor, “Novel topological quasi-soliton solutions for the nonlinear cubic-quintic equation model,” Proc. SPIE 4271, 292–302 (2001).
[Crossref]

Berry, M. V.

M. V. Berry and M. R. Dennis, “Quantum cores of optical phase singularities,” J. Opt. A, Pure Appl. Opt. 6(5), S178–S180 (2004).
[Crossref]

Birnbaum, Z.

Z. Birnbaum and B. A. Malomed, “Families of spatial solitons in a two-channel waveguide with the cubic-quintic nonlinearity,” Physica D 237(24), 3252–3262 (2008).
[Crossref]

Boudebs, G.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219(1-6), 427–433 (2003).
[Crossref]

Caplan, R. M.

R. M. Caplan, R. Carretero-González, P. G. Kevrekidis, and B. A. Malomed, “Existence, stability, and scattering of bright vortices in the cubic–quintic nonlinear Schrödinger equation,” Math. Comput. Simul. 82(7), 1150–1171 (2012).
[Crossref]

Caputo, J. G.

F. K. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length,” Phys. Rev. A 67(1), 013605 (2003).
[Crossref]

Cardoso, W. B.

A. T. Avelar, D. Bazeia, and W. B. Cardoso, “Solitons with cubic and quintic nonlinearities modulated in space and time,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(2), 025602 (2009).
[Crossref] [PubMed]

Carretero-González, R.

R. M. Caplan, R. Carretero-González, P. G. Kevrekidis, and B. A. Malomed, “Existence, stability, and scattering of bright vortices in the cubic–quintic nonlinear Schrödinger equation,” Math. Comput. Simul. 82(7), 1150–1171 (2012).
[Crossref]

Chen, G.

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation,” Phys. Rev. Lett. 101(12), 123904 (2008).
[Crossref] [PubMed]

Chen, J. C.

J. C. Chen, X. F. Zhang, B. Li, and Y. Chen, “Exact solutions to the two-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrödinger equation with an external potential,” Chin. Phys. Lett. 29, 220 (2012).

Chen, Y.

J. C. Chen, X. F. Zhang, B. Li, and Y. Chen, “Exact solutions to the two-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrödinger equation with an external potential,” Chin. Phys. Lett. 29, 220 (2012).

Chen, Z.

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004).
[Crossref] [PubMed]

Cherukulappurath, S.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219(1-6), 427–433 (2003).
[Crossref]

Christodoulides, D. N.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92(16), 163902 (2004).
[Crossref] [PubMed]

Cuevas, J.

J. Belmonte-Beitia and J. Cuevas, “Solitons for the cubic-quintic nonlinear Schrodinger equation with time- and space-modulated coefficients,” J. Phys. A 42, 11 (2009).

Dennis, M. R.

M. V. Berry and M. R. Dennis, “Quantum cores of optical phase singularities,” J. Opt. A, Pure Appl. Opt. 6(5), S178–S180 (2004).
[Crossref]

Desyatnikov, A. S.

A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. 95(20), 203904 (2005).
[Crossref] [PubMed]

Hao, R.

R. Hao, L. Li, Z. Li, R. Yang, and G. Zhou, “A new approach to exact soliton solutions and soliton interaction for the nonlinear Schrödinger equation with variable coefficients,” Opt. Commun. 245, 383 (2005).
[Crossref]

Hasegawa, A.

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous matter-wave solitons near the Feshbach resonance,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 81, 023610 (2010).

He, J.-R.

J.-R. He, L. Yi, and H.-M. Li, “Self-similar propagation and asymptotic optical waves in nonlinear waveguides,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(1), 013202 (2014).
[Crossref] [PubMed]

Hu, B.

Kartashov, Y. V.

Kevrekidis, P. G.

R. M. Caplan, R. Carretero-González, P. G. Kevrekidis, and B. A. Malomed, “Existence, stability, and scattering of bright vortices in the cubic–quintic nonlinear Schrödinger equation,” Math. Comput. Simul. 82(7), 1150–1171 (2012).
[Crossref]

Kivshar, Y. S.

D. Mihalache, D. Mazilu, F. Lederer, and Y. S. Kivshar, “Spatiotemporal surface solitons in two-dimensional photonic lattices,” Opt. Lett. 32(21), 3173–3175 (2007).
[Crossref] [PubMed]

A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. 95(20), 203904 (2005).
[Crossref] [PubMed]

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004).
[Crossref] [PubMed]

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61(4), 763–915 (1989).
[Crossref]

Kraenkel, R. A.

F. K. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length,” Phys. Rev. A 67(1), 013605 (2003).
[Crossref]

Kruskal, M. D.

N. J. Zabusky and M. D. Kruskal, “Interactions of solitons in a collisionless plasma and the recurrence of initial states,” Phys. Rev. Lett. 15(6), 240–243 (1965).
[Crossref]

Lakoba, T. I.

J. K. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118(2), 153 (2007).
[Crossref]

Leblond, H.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability limits for three-dimensional vortex solitons in the Ginzburg-Landau equation with the cubic-quintic nonlinearity,” Phys. Rev. A 76(4), 0453803 (2007).
[Crossref]

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219(1-6), 427–433 (2003).
[Crossref]

Lederer, F.

D. Mihalache, D. Mazilu, F. Lederer, and Y. S. Kivshar, “Spatiotemporal surface solitons in two-dimensional photonic lattices,” Opt. Lett. 32(21), 3173–3175 (2007).
[Crossref] [PubMed]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability limits for three-dimensional vortex solitons in the Ginzburg-Landau equation with the cubic-quintic nonlinearity,” Phys. Rev. A 76(4), 0453803 (2007).
[Crossref]

Li, B.

J. C. Chen, X. F. Zhang, B. Li, and Y. Chen, “Exact solutions to the two-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrödinger equation with an external potential,” Chin. Phys. Lett. 29, 220 (2012).

Li, H.-M.

J.-R. He, L. Yi, and H.-M. Li, “Self-similar propagation and asymptotic optical waves in nonlinear waveguides,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(1), 013202 (2014).
[Crossref] [PubMed]

Li, L.

R. Hao, L. Li, Z. Li, R. Yang, and G. Zhou, “A new approach to exact soliton solutions and soliton interaction for the nonlinear Schrödinger equation with variable coefficients,” Opt. Commun. 245, 383 (2005).
[Crossref]

Li, Z.

R. Hao, L. Li, Z. Li, R. Yang, and G. Zhou, “A new approach to exact soliton solutions and soliton interaction for the nonlinear Schrödinger equation with variable coefficients,” Opt. Commun. 245, 383 (2005).
[Crossref]

Liang, J. C.

Makasyuk, I.

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004).
[Crossref] [PubMed]

Malomed, B. A.

R. M. Caplan, R. Carretero-González, P. G. Kevrekidis, and B. A. Malomed, “Existence, stability, and scattering of bright vortices in the cubic–quintic nonlinear Schrödinger equation,” Math. Comput. Simul. 82(7), 1150–1171 (2012).
[Crossref]

T. Mayteevarunyoo, B. A. Malomed, and A. Roeksabutr, “Solitons and vortices in nonlinear two-dimensional photonic crystals of the Kronig-Penney type,” Opt. Express 19(18), 17834–17851 (2011).
[Crossref] [PubMed]

Z. Birnbaum and B. A. Malomed, “Families of spatial solitons in a two-channel waveguide with the cubic-quintic nonlinearity,” Physica D 237(24), 3252–3262 (2008).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability limits for three-dimensional vortex solitons in the Ginzburg-Landau equation with the cubic-quintic nonlinearity,” Phys. Rev. A 76(4), 0453803 (2007).
[Crossref]

F. K. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length,” Phys. Rev. A 67(1), 013605 (2003).
[Crossref]

B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in periodic potentials,” Europhys. Lett. 63(5), 642–648 (2003).
[Crossref]

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61(4), 763–915 (1989).
[Crossref]

Martin, H.

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004).
[Crossref] [PubMed]

Mayteevarunyoo, T.

Mazilu, D.

D. Mihalache, D. Mazilu, F. Lederer, and Y. S. Kivshar, “Spatiotemporal surface solitons in two-dimensional photonic lattices,” Opt. Lett. 32(21), 3173–3175 (2007).
[Crossref] [PubMed]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability limits for three-dimensional vortex solitons in the Ginzburg-Landau equation with the cubic-quintic nonlinearity,” Phys. Rev. A 76(4), 0453803 (2007).
[Crossref]

Meier, J.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92(16), 163902 (2004).
[Crossref] [PubMed]

Melchor, G. M.

V. N. Serkin, T. L. Belyaeva, I. V. Alexandrov, and G. M. Melchor, “Novel topological quasi-soliton solutions for the nonlinear cubic-quintic equation model,” Proc. SPIE 4271, 292–302 (2001).
[Crossref]

Mihalache, D.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability limits for three-dimensional vortex solitons in the Ginzburg-Landau equation with the cubic-quintic nonlinearity,” Phys. Rev. A 76(4), 0453803 (2007).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, and Y. S. Kivshar, “Spatiotemporal surface solitons in two-dimensional photonic lattices,” Opt. Lett. 32(21), 3173–3175 (2007).
[Crossref] [PubMed]

Morandotti, R.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92(16), 163902 (2004).
[Crossref] [PubMed]

Mullin, W. J.

W. J. Mullin and A. R. Sakhel, “Generalized Bose-Einstein condensation,” J. Low Temp. Phys. 166(3-4), 125–150 (2012).
[Crossref]

Neshev, D. N.

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004).
[Crossref] [PubMed]

Ostrovskaya, E. A.

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004).
[Crossref] [PubMed]

Pérez-García, V. M.

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98(6), 064102 (2007).
[Crossref] [PubMed]

Petrovic, N.

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation,” Phys. Rev. Lett. 101(12), 123904 (2008).
[Crossref] [PubMed]

Petrovic, N. Z.

S. L. Xu, G. P. Zhou, N. Z. Petrović, and M. R. Belić, “Two-dimensional dark solitons in diffusive nonlocal nonlinear media,” J. Opt. 17, 105605 (2015).
[Crossref]

Roeksabutr, A.

Sakhel, A. R.

W. J. Mullin and A. R. Sakhel, “Generalized Bose-Einstein condensation,” J. Low Temp. Phys. 166(3-4), 125–150 (2012).
[Crossref]

Salamo, G.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92(16), 163902 (2004).
[Crossref] [PubMed]

Salerno, M.

B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in periodic potentials,” Europhys. Lett. 63(5), 642–648 (2003).
[Crossref]

Sanchez, F.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219(1-6), 427–433 (2003).
[Crossref]

Serkin, V. N.

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous matter-wave solitons near the Feshbach resonance,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 81, 023610 (2010).

V. N. Serkin, T. L. Belyaeva, I. V. Alexandrov, and G. M. Melchor, “Novel topological quasi-soliton solutions for the nonlinear cubic-quintic equation model,” Proc. SPIE 4271, 292–302 (2001).
[Crossref]

Shukla, P. K.

X. Y. Tang and P. K. Shukla, “Solution of the one-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrödinger equation with an external potential,” Phys. Rev. A 76(1), 013612 (2007).
[Crossref]

Silberberg, Y.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92(16), 163902 (2004).
[Crossref] [PubMed]

Smektala, F.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219(1-6), 427–433 (2003).
[Crossref]

Sorel, M.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92(16), 163902 (2004).
[Crossref] [PubMed]

Stegeman, G. I.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92(16), 163902 (2004).
[Crossref] [PubMed]

Sukhorukov, A. A.

A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. 95(20), 203904 (2005).
[Crossref] [PubMed]

Tang, X. Y.

X. Y. Tang and P. K. Shukla, “Solution of the one-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrödinger equation with an external potential,” Phys. Rev. A 76(1), 013612 (2007).
[Crossref]

Torner, L.

Torres, P. J.

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98(6), 064102 (2007).
[Crossref] [PubMed]

Troles, J.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219(1-6), 427–433 (2003).
[Crossref]

Vekslerchik, V.

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98(6), 064102 (2007).
[Crossref] [PubMed]

Xie, R. H.

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation,” Phys. Rev. Lett. 101(12), 123904 (2008).
[Crossref] [PubMed]

Xu, S. L.

S. L. Xu, G. P. Zhou, N. Z. Petrović, and M. R. Belić, “Two-dimensional dark solitons in diffusive nonlocal nonlinear media,” J. Opt. 17, 105605 (2015).
[Crossref]

S. L. Xu, J. C. Liang, and L. Yi, “Self-similar solitary waves in Bessel optical lattices,” J. Opt. Soc. Am. B 27(1), 99 (2010).
[Crossref]

Xu, S.-L.

S.-L. Xu and M. R. Belic, “Light bullets in three-dimensional complex Ginzburg-Landau equation with modulated Kummer-Gauss photonic lattice,” EPL 108(3), 34001 (2014).
[Crossref]

Yang, H.

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92(16), 163902 (2004).
[Crossref] [PubMed]

Yang, J. K.

J. K. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118(2), 153 (2007).
[Crossref]

Yang, R.

R. Hao, L. Li, Z. Li, R. Yang, and G. Zhou, “A new approach to exact soliton solutions and soliton interaction for the nonlinear Schrödinger equation with variable coefficients,” Opt. Commun. 245, 383 (2005).
[Crossref]

Ye, F.

Yi, L.

J.-R. He, L. Yi, and H.-M. Li, “Self-similar propagation and asymptotic optical waves in nonlinear waveguides,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(1), 013202 (2014).
[Crossref] [PubMed]

S. L. Xu, J. C. Liang, and L. Yi, “Self-similar solitary waves in Bessel optical lattices,” J. Opt. Soc. Am. B 27(1), 99 (2010).
[Crossref]

Zabusky, N. J.

N. J. Zabusky and M. D. Kruskal, “Interactions of solitons in a collisionless plasma and the recurrence of initial states,” Phys. Rev. Lett. 15(6), 240–243 (1965).
[Crossref]

Zhang, X. F.

J. C. Chen, X. F. Zhang, B. Li, and Y. Chen, “Exact solutions to the two-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrödinger equation with an external potential,” Chin. Phys. Lett. 29, 220 (2012).

Zhong, W. P.

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation,” Phys. Rev. Lett. 101(12), 123904 (2008).
[Crossref] [PubMed]

Zhou, G.

R. Hao, L. Li, Z. Li, R. Yang, and G. Zhou, “A new approach to exact soliton solutions and soliton interaction for the nonlinear Schrödinger equation with variable coefficients,” Opt. Commun. 245, 383 (2005).
[Crossref]

Zhou, G. P.

S. L. Xu, G. P. Zhou, N. Z. Petrović, and M. R. Belić, “Two-dimensional dark solitons in diffusive nonlocal nonlinear media,” J. Opt. 17, 105605 (2015).
[Crossref]

Chin. Phys. Lett. (1)

J. C. Chen, X. F. Zhang, B. Li, and Y. Chen, “Exact solutions to the two-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrödinger equation with an external potential,” Chin. Phys. Lett. 29, 220 (2012).

EPL (1)

S.-L. Xu and M. R. Belic, “Light bullets in three-dimensional complex Ginzburg-Landau equation with modulated Kummer-Gauss photonic lattice,” EPL 108(3), 34001 (2014).
[Crossref]

Europhys. Lett. (1)

B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in periodic potentials,” Europhys. Lett. 63(5), 642–648 (2003).
[Crossref]

J. Low Temp. Phys. (1)

W. J. Mullin and A. R. Sakhel, “Generalized Bose-Einstein condensation,” J. Low Temp. Phys. 166(3-4), 125–150 (2012).
[Crossref]

J. Opt. (1)

S. L. Xu, G. P. Zhou, N. Z. Petrović, and M. R. Belić, “Two-dimensional dark solitons in diffusive nonlocal nonlinear media,” J. Opt. 17, 105605 (2015).
[Crossref]

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

M. V. Berry and M. R. Dennis, “Quantum cores of optical phase singularities,” J. Opt. A, Pure Appl. Opt. 6(5), S178–S180 (2004).
[Crossref]

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

J. Phys. A (1)

J. Belmonte-Beitia and J. Cuevas, “Solitons for the cubic-quintic nonlinear Schrodinger equation with time- and space-modulated coefficients,” J. Phys. A 42, 11 (2009).

Math. Comput. Simul. (1)

R. M. Caplan, R. Carretero-González, P. G. Kevrekidis, and B. A. Malomed, “Existence, stability, and scattering of bright vortices in the cubic–quintic nonlinear Schrödinger equation,” Math. Comput. Simul. 82(7), 1150–1171 (2012).
[Crossref]

Opt. Commun. (2)

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219(1-6), 427–433 (2003).
[Crossref]

R. Hao, L. Li, Z. Li, R. Yang, and G. Zhou, “A new approach to exact soliton solutions and soliton interaction for the nonlinear Schrödinger equation with variable coefficients,” Opt. Commun. 245, 383 (2005).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. A (3)

X. Y. Tang and P. K. Shukla, “Solution of the one-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrödinger equation with an external potential,” Phys. Rev. A 76(1), 013612 (2007).
[Crossref]

F. K. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length,” Phys. Rev. A 67(1), 013605 (2003).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability limits for three-dimensional vortex solitons in the Ginzburg-Landau equation with the cubic-quintic nonlinearity,” Phys. Rev. A 76(4), 0453803 (2007).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (3)

J.-R. He, L. Yi, and H.-M. Li, “Self-similar propagation and asymptotic optical waves in nonlinear waveguides,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(1), 013202 (2014).
[Crossref] [PubMed]

A. T. Avelar, D. Bazeia, and W. B. Cardoso, “Solitons with cubic and quintic nonlinearities modulated in space and time,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(2), 025602 (2009).
[Crossref] [PubMed]

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous matter-wave solitons near the Feshbach resonance,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 81, 023610 (2010).

Phys. Rev. Lett. (6)

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98(6), 064102 (2007).
[Crossref] [PubMed]

A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. 95(20), 203904 (2005).
[Crossref] [PubMed]

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation,” Phys. Rev. Lett. 101(12), 123904 (2008).
[Crossref] [PubMed]

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92(16), 163902 (2004).
[Crossref] [PubMed]

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004).
[Crossref] [PubMed]

N. J. Zabusky and M. D. Kruskal, “Interactions of solitons in a collisionless plasma and the recurrence of initial states,” Phys. Rev. Lett. 15(6), 240–243 (1965).
[Crossref]

Physica D (1)

Z. Birnbaum and B. A. Malomed, “Families of spatial solitons in a two-channel waveguide with the cubic-quintic nonlinearity,” Physica D 237(24), 3252–3262 (2008).
[Crossref]

Proc. SPIE (1)

V. N. Serkin, T. L. Belyaeva, I. V. Alexandrov, and G. M. Melchor, “Novel topological quasi-soliton solutions for the nonlinear cubic-quintic equation model,” Proc. SPIE 4271, 292–302 (2001).
[Crossref]

Rev. Mod. Phys. (1)

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61(4), 763–915 (1989).
[Crossref]

Stud. Appl. Math. (1)

J. K. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118(2), 153 (2007).
[Crossref]

Other (3)

J. K. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM, 2010).

D. Zwillinger, Handbook of Differential Equations, 3rd ed. (Academic Press, 1997).

C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, (Springer-Verlag, 2000)

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Figures (8)

Fig. 1
Fig. 1 Distributions of the nonlinearity coefficients χ 1 ( r ) , χ 2 ( r ) and the external potential V ( r ) for the resonance soliton from Eq. (9). Parameters: w 0 = a 0 = a 1 = 1 , q = 0.2 , m 0 = 0.1 , m = 0 , n = 1 and s = 0.1 .
Fig. 2
Fig. 2 Intensity profiles (top row) of resonance solitons for the cubic-quintic nonlinearities and potentials at the propagation distance z = 50 for E=0 and different m; Phase distributions (middle row); Linear-stability spectra (bottom row). The parameters are: n = 2 , q = 0.5 , w 0 = 1 and m = 1 , 2 , 3 from left to right.
Fig. 3
Fig. 3 Same as Fig. 2, but for E = 1 , and a 1 = 1 .
Fig. 4
Fig. 4 Same as Fig. 2, but for m = 1 , n = 0 , 1 , 2 from left to right.
Fig. 5
Fig. 5 Same as Fig. 4, but for E = 1 , and a 1 = 1 .
Fig. 6
Fig. 6 Same as Fig. 2, but for n = 1 , m = 2 , and q = 0.1 , 0.5 , 1 from left to right.
Fig. 7
Fig. 7 Same as Fig. 5, but for E = 1 .
Fig. 8
Fig. 8 Comparison of analytical (the first column) and numerical (the second and the third columns) intensity distribution contour plots at different distances z = 10 , 100 , 160 in the x-y plane and with a white noise of variance σ = 0.06 added to the numerical input. The parameters are E = 0 , q = 0 , n = m = 0 (the first row), n = m = 1 (the second row), n = m = 2 (the third row).

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

i ψ z + 1 2 ψ + χ 1 ( z , r ) | ψ | 2 ψ + χ 2 ( z , r ) | ψ | 4 ψ + V ( z , r ) ψ = 0 ,
i u z + 1 2 ( 2 u r 2 + 1 r u r m 2 r 2 u ) + χ 1 ' ( z , r ) | u | 2 u + χ 2 ' ( z , r ) | u | 4 u + V ( z , r ) u = 0.
U R R = E U + g 0 | U | 2 U + G 0 | U | 4 U ,
U ( R ) = 3 s n ( μ R , m 0 ) 3 a 0 [ 3 ( m 0 2 + 1 ) s n 2 ( μ R , m 0 ) ] .
U ( R ) = s n ( μ R , m 0 ) a 0 + a 1 d n 2 ( μ R , m 0 ) ,
μ = E ( a 0 + a 1 ) / 3 ( 2 m 0 2 a 1 m 0 2 a 0 a 0 a 1 ) ,
g 0 = 2 E m 0 2 ( a 0 2 2 a 1 a 0 m 0 2 a 1 2 + a 1 2 m 0 2 ) / ( m 0 2 a 0 2 m 0 2 a 1 + a 0 + a 1 ) ,
G 0 = 3 E a 1 a 0 m 0 4 ( a 0 + a 1 a 1 2 m 0 2 ) / ( m 0 2 a 0 2 m 0 2 a 1 + a 0 + a 1 ) ,
A z + 1 2 ( A 2 Θ r 2 + 2 A r Θ r + A r Θ r ) = 0 , R z + R r Θ r = 0
A Θ z + 1 2 [ 2 A r 2 A ( Θ r ) 2 + 1 r A r A m 2 r 2 ] + A V E A ( R r ) 2 = 0 ,
2 A r R r + A 2 R r 2 + A r R r = 0 , χ 1 ' A 2 = g 0 2 R r 2 , χ 2 ' A 4 = G 0 2 R r 2
θ d 2 F d θ 2 + d F d θ ( θ w 3 4 d 2 w d z 2 + m 2 4 θ ) F w 2 F 2 d b d z w 2 2 [ V + E ( R r ) 2 ] F = 0.
θ d 2 f d θ 2 + ( m + 1 θ ) d f d θ n f = 0 ,
ψ ( z , r , φ ) = [ cos ( m φ ) + i q sin ( m φ ) ] k 1 w 0 ( r w 0 ) m e r 2 2 w 0 2 S n m ( r 2 w 0 2 ) U [ R ( r ) ] e i [ b 0 ( 2 n + m + 1 ) w 0 2 z ] .
ψ ( x , y , z ) = e i λ z { ψ ¯ ( x , y ) + ε [ g ( x , y ) + h ( x , y ) ] e i δ z } ,
( L + 0 0 L ) ( g h ) = δ ( g h ) ,
L + = 1 2 ( x x + y y ) + 3 χ 1 ψ ¯ 2 + 5 χ 2 ψ ¯ 4 + V λ , L = 1 2 ( x x + y y ) + χ 1 ψ ¯ 2 + χ 2 ψ ¯ 4 + V λ

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