Abstract

We introduce a numerical variational method based on the Rayleigh-Ritz optimization principle for predicting two-dimensional self-trapped beams in nonlinear media. This technique overcomes the limitation of the traditional variational approximation in performing analytical Lagrangian integration and differentiation. Approximate soliton solutions of a generalized nonlinear Schrödinger equation are obtained, demonstrating robustness of the beams of various types (fundamental, vortices, multipoles, azimuthons) in the course of their propagation. The algorithm offers possibilities to produce more sophisticated soliton profiles in general nonlinear models.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. Y. S. Kivshar, G. P. Agrawal, and G. P. Agrawal, eds., Index (Academic Press, Burlington, 2003).
  2. Z. Chen, M. Segev, and D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Reports on Prog. Phys. 75, 086401 (2012).
    [Crossref]
  3. C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the korteweg-devries equation,” Phys. Rev. Lett. 19, 1095–1097 (1967).
    [Crossref]
  4. H. D. Wahlquist and F. B. Estabrook, “Bäcklund transformation for solutions of the korteweg-de vries equation,” Phys. Rev. Lett. 31, 1386–1390 (1973).
    [Crossref]
  5. J. Yang, “Newton-conjugate-gradient methods for solitary wave computations,” J. Comput. Phys. 228, 7007–7024 (2009).
    [Crossref]
  6. O. V. Matusevich and V. A. Trofimov, “Iterative method for finding the eigenfunctions of a system of two Schrödinger equations with combined nonlinearity,” Comput. Math. Math. Phys. 48, 677–687 (2008).
    [Crossref]
  7. O. V. Matusevich and V. a. Trofimov, “A numerical method for calculating solitons of the nonlinear Schödinger equation in the axially symmetric case,” Mosc. Univ. Comput. Math. Cybern. 33, 117–126 (2009).
    [Crossref]
  8. V. Trofimov, T. Lysak, O. Matusevich, and S. Lan, “Parameter control of optical soliton in one-dimensional photonic crystal,” Math. Model. Analysis 15, 517–532 (2010).
    [Crossref]
  9. D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
    [Crossref]
  10. A. Bondeson, M. Lisak, and D. Anderson, “Soliton perturbations: a variational principle for the soliton parameters,” Phys. Scripta 20, 479 (1979).
    [Crossref]
  11. D. Anderson, M. Lisak, and T. Reichel, “Asymptotic propagation properties of pulses in a soliton-based optical-fiber communication system,” JOSA B 5, 207–210 (1988).
    [Crossref]
  12. B. A. Malomed, “Variational methods in nonlinear fiber optics and related fields,” Prog. optics 43, 71–194 (2002).
    [Crossref]
  13. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton shape and mobility control in optical lattices,” Prog. Opt. 52, 63–148 (2009).
    [Crossref]
  14. V. Skarka and N. Aleksić, “Stability criterion for dissipative soliton solutions of the one-, two-, and three-dimensional complex cubic-quintic ginzburg-landau equations,” Phys. review letters 96, 013903 (2006).
    [Crossref]
  15. A. Sahoo, S. Roy, and G. P. Agrawal, “Perturbed dissipative solitons: A variational approach,” Phys. Rev. A 96, 1–8 (2017).
    [Crossref]
  16. T. Busch, J. Cirac, V. Perez-Garcia, and P. Zoller, “Stability and collective excitations of a two-component bose-einstein condensed gas: A moment approach,” Phys. Rev. A 56, 2978 (1997).
    [Crossref]
  17. D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L. C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in bessel optical lattices,” Phys. Rev. Lett. 95, 10703–10710 (2005).
    [Crossref]
  18. S. Hu, H. Chen, and W. Hu, “A variational solution to solitons in parity-time symmetric optical lattices,” Opt. Commun. 320, 60–67 (2014).
    [Crossref]
  19. A. B. Aceves and C. De Angelis, “Spatiotemporal pulse dynamics in a periodic nonlinear waveguide,” Opt. Lett. 18, 110–112 (1993).
    [Crossref] [PubMed]
  20. Y. V. Kartashov, B. A. Malomed, Y. Shnir, and L. Torner, “Twisted toroidal vortex solitons in inhomogeneous media with repulsive nonlinearity,” Phys. review letters 113, 264101 (2014).
    [Crossref]
  21. V. M. Pérez-García, V. V. Konotop, and J. J. García-Ripoll, “Dynamics of quasicollapse in nonlinear schrödinger systems with nonlocal interactions,” Phys. Rev. E 62, 4300 (2000).
    [Crossref]
  22. Z. Dai, Z. Yang, X. Ling, S. Zhang, Z. Pang, and X. Li, “Tripole-mode and quadrupole-mode solitons in media with a spatial exponential-decay nonlocality,” Sci. Rep. 7, 1–13 (2017).
  23. D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana 57, 917–936 (2001).
    [Crossref]
  24. M. Chiofalo, S. Succi, and M. Tosi, “Ground state of trapped interacting bose-einstein condensates by an explicit imaginary-time algorithm,” Phys. Rev. E 62, 7438 (2000).
    [Crossref]
  25. W. Bao and Q. Du, “Computing the ground state solution of bose–einstein condensates by a normalized gradient flow,” SIAM J. on Sci. Comput. 25, 1674–1697 (2004).
    [Crossref]
  26. A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: Spatially modulated vortex solitons,” Phys. Rev. Lett. 95, 203904 (2005).
    [Crossref] [PubMed]
  27. S. Lopez-Aguayo and J. C. Gutierrez-Vega, “Elliptically modulated self-trapped singular beams in nonlocal nonlinear media: ellipticons,” Opt. Express 15, 18326–18338 (2007).
    [Crossref] [PubMed]
  28. K. Dimitrevski, E. Reimhult, E. Svensson, A. Öhgren, D. Anderson, A. Berntson, M. Lisak, and M. L. Quiroga-Teixeiro, “Analysis of stable self-trapping of laser beams in cubic-quintic nonlinear media,” Phys. Lett. A 248, 369–376 (1998).
    [Crossref]
  29. B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B: Quantum Semiclassical Opt. 7, R53 (2005).
    [Crossref]

2017 (2)

A. Sahoo, S. Roy, and G. P. Agrawal, “Perturbed dissipative solitons: A variational approach,” Phys. Rev. A 96, 1–8 (2017).
[Crossref]

Z. Dai, Z. Yang, X. Ling, S. Zhang, Z. Pang, and X. Li, “Tripole-mode and quadrupole-mode solitons in media with a spatial exponential-decay nonlocality,” Sci. Rep. 7, 1–13 (2017).

2014 (2)

Y. V. Kartashov, B. A. Malomed, Y. Shnir, and L. Torner, “Twisted toroidal vortex solitons in inhomogeneous media with repulsive nonlinearity,” Phys. review letters 113, 264101 (2014).
[Crossref]

S. Hu, H. Chen, and W. Hu, “A variational solution to solitons in parity-time symmetric optical lattices,” Opt. Commun. 320, 60–67 (2014).
[Crossref]

2012 (1)

Z. Chen, M. Segev, and D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Reports on Prog. Phys. 75, 086401 (2012).
[Crossref]

2010 (1)

V. Trofimov, T. Lysak, O. Matusevich, and S. Lan, “Parameter control of optical soliton in one-dimensional photonic crystal,” Math. Model. Analysis 15, 517–532 (2010).
[Crossref]

2009 (3)

O. V. Matusevich and V. a. Trofimov, “A numerical method for calculating solitons of the nonlinear Schödinger equation in the axially symmetric case,” Mosc. Univ. Comput. Math. Cybern. 33, 117–126 (2009).
[Crossref]

J. Yang, “Newton-conjugate-gradient methods for solitary wave computations,” J. Comput. Phys. 228, 7007–7024 (2009).
[Crossref]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton shape and mobility control in optical lattices,” Prog. Opt. 52, 63–148 (2009).
[Crossref]

2008 (1)

O. V. Matusevich and V. A. Trofimov, “Iterative method for finding the eigenfunctions of a system of two Schrödinger equations with combined nonlinearity,” Comput. Math. Math. Phys. 48, 677–687 (2008).
[Crossref]

2007 (1)

2006 (1)

V. Skarka and N. Aleksić, “Stability criterion for dissipative soliton solutions of the one-, two-, and three-dimensional complex cubic-quintic ginzburg-landau equations,” Phys. review letters 96, 013903 (2006).
[Crossref]

2005 (3)

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L. C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in bessel optical lattices,” Phys. Rev. Lett. 95, 10703–10710 (2005).
[Crossref]

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

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B: Quantum Semiclassical Opt. 7, R53 (2005).
[Crossref]

2004 (1)

W. Bao and Q. Du, “Computing the ground state solution of bose–einstein condensates by a normalized gradient flow,” SIAM J. on Sci. Comput. 25, 1674–1697 (2004).
[Crossref]

2002 (1)

B. A. Malomed, “Variational methods in nonlinear fiber optics and related fields,” Prog. optics 43, 71–194 (2002).
[Crossref]

2001 (1)

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana 57, 917–936 (2001).
[Crossref]

2000 (2)

M. Chiofalo, S. Succi, and M. Tosi, “Ground state of trapped interacting bose-einstein condensates by an explicit imaginary-time algorithm,” Phys. Rev. E 62, 7438 (2000).
[Crossref]

V. M. Pérez-García, V. V. Konotop, and J. J. García-Ripoll, “Dynamics of quasicollapse in nonlinear schrödinger systems with nonlocal interactions,” Phys. Rev. E 62, 4300 (2000).
[Crossref]

1998 (1)

K. Dimitrevski, E. Reimhult, E. Svensson, A. Öhgren, D. Anderson, A. Berntson, M. Lisak, and M. L. Quiroga-Teixeiro, “Analysis of stable self-trapping of laser beams in cubic-quintic nonlinear media,” Phys. Lett. A 248, 369–376 (1998).
[Crossref]

1997 (1)

T. Busch, J. Cirac, V. Perez-Garcia, and P. Zoller, “Stability and collective excitations of a two-component bose-einstein condensed gas: A moment approach,” Phys. Rev. A 56, 2978 (1997).
[Crossref]

1993 (1)

1988 (1)

D. Anderson, M. Lisak, and T. Reichel, “Asymptotic propagation properties of pulses in a soliton-based optical-fiber communication system,” JOSA B 5, 207–210 (1988).
[Crossref]

1983 (1)

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[Crossref]

1979 (1)

A. Bondeson, M. Lisak, and D. Anderson, “Soliton perturbations: a variational principle for the soliton parameters,” Phys. Scripta 20, 479 (1979).
[Crossref]

1973 (1)

H. D. Wahlquist and F. B. Estabrook, “Bäcklund transformation for solutions of the korteweg-de vries equation,” Phys. Rev. Lett. 31, 1386–1390 (1973).
[Crossref]

1967 (1)

C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the korteweg-devries equation,” Phys. Rev. Lett. 19, 1095–1097 (1967).
[Crossref]

Aceves, A. B.

Agrawal, G. P.

A. Sahoo, S. Roy, and G. P. Agrawal, “Perturbed dissipative solitons: A variational approach,” Phys. Rev. A 96, 1–8 (2017).
[Crossref]

Aleksic, N.

V. Skarka and N. Aleksić, “Stability criterion for dissipative soliton solutions of the one-, two-, and three-dimensional complex cubic-quintic ginzburg-landau equations,” Phys. review letters 96, 013903 (2006).
[Crossref]

Anderson, D.

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana 57, 917–936 (2001).
[Crossref]

K. Dimitrevski, E. Reimhult, E. Svensson, A. Öhgren, D. Anderson, A. Berntson, M. Lisak, and M. L. Quiroga-Teixeiro, “Analysis of stable self-trapping of laser beams in cubic-quintic nonlinear media,” Phys. Lett. A 248, 369–376 (1998).
[Crossref]

D. Anderson, M. Lisak, and T. Reichel, “Asymptotic propagation properties of pulses in a soliton-based optical-fiber communication system,” JOSA B 5, 207–210 (1988).
[Crossref]

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[Crossref]

A. Bondeson, M. Lisak, and D. Anderson, “Soliton perturbations: a variational principle for the soliton parameters,” Phys. Scripta 20, 479 (1979).
[Crossref]

Bao, W.

W. Bao and Q. Du, “Computing the ground state solution of bose–einstein condensates by a normalized gradient flow,” SIAM J. on Sci. Comput. 25, 1674–1697 (2004).
[Crossref]

Berntson, A.

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana 57, 917–936 (2001).
[Crossref]

K. Dimitrevski, E. Reimhult, E. Svensson, A. Öhgren, D. Anderson, A. Berntson, M. Lisak, and M. L. Quiroga-Teixeiro, “Analysis of stable self-trapping of laser beams in cubic-quintic nonlinear media,” Phys. Lett. A 248, 369–376 (1998).
[Crossref]

Bondeson, A.

A. Bondeson, M. Lisak, and D. Anderson, “Soliton perturbations: a variational principle for the soliton parameters,” Phys. Scripta 20, 479 (1979).
[Crossref]

Busch, T.

T. Busch, J. Cirac, V. Perez-Garcia, and P. Zoller, “Stability and collective excitations of a two-component bose-einstein condensed gas: A moment approach,” Phys. Rev. A 56, 2978 (1997).
[Crossref]

Chen, H.

S. Hu, H. Chen, and W. Hu, “A variational solution to solitons in parity-time symmetric optical lattices,” Opt. Commun. 320, 60–67 (2014).
[Crossref]

Chen, Z.

Z. Chen, M. Segev, and D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Reports on Prog. Phys. 75, 086401 (2012).
[Crossref]

Chiofalo, M.

M. Chiofalo, S. Succi, and M. Tosi, “Ground state of trapped interacting bose-einstein condensates by an explicit imaginary-time algorithm,” Phys. Rev. E 62, 7438 (2000).
[Crossref]

Christodoulides, D. N.

Z. Chen, M. Segev, and D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Reports on Prog. Phys. 75, 086401 (2012).
[Crossref]

Cirac, J.

T. Busch, J. Cirac, V. Perez-Garcia, and P. Zoller, “Stability and collective excitations of a two-component bose-einstein condensed gas: A moment approach,” Phys. Rev. A 56, 2978 (1997).
[Crossref]

Crasovan, L. C.

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L. C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in bessel optical lattices,” Phys. Rev. Lett. 95, 10703–10710 (2005).
[Crossref]

Dai, Z.

Z. Dai, Z. Yang, X. Ling, S. Zhang, Z. Pang, and X. Li, “Tripole-mode and quadrupole-mode solitons in media with a spatial exponential-decay nonlocality,” Sci. Rep. 7, 1–13 (2017).

De Angelis, C.

Desyatnikov, A. S.

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

Dimitrevski, K.

K. Dimitrevski, E. Reimhult, E. Svensson, A. Öhgren, D. Anderson, A. Berntson, M. Lisak, and M. L. Quiroga-Teixeiro, “Analysis of stable self-trapping of laser beams in cubic-quintic nonlinear media,” Phys. Lett. A 248, 369–376 (1998).
[Crossref]

Du, Q.

W. Bao and Q. Du, “Computing the ground state solution of bose–einstein condensates by a normalized gradient flow,” SIAM J. on Sci. Comput. 25, 1674–1697 (2004).
[Crossref]

Estabrook, F. B.

H. D. Wahlquist and F. B. Estabrook, “Bäcklund transformation for solutions of the korteweg-de vries equation,” Phys. Rev. Lett. 31, 1386–1390 (1973).
[Crossref]

García-Ripoll, J. J.

V. M. Pérez-García, V. V. Konotop, and J. J. García-Ripoll, “Dynamics of quasicollapse in nonlinear schrödinger systems with nonlocal interactions,” Phys. Rev. E 62, 4300 (2000).
[Crossref]

Gardner, C. S.

C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the korteweg-devries equation,” Phys. Rev. Lett. 19, 1095–1097 (1967).
[Crossref]

Greene, J. M.

C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the korteweg-devries equation,” Phys. Rev. Lett. 19, 1095–1097 (1967).
[Crossref]

Gutierrez-Vega, J. C.

Hu, S.

S. Hu, H. Chen, and W. Hu, “A variational solution to solitons in parity-time symmetric optical lattices,” Opt. Commun. 320, 60–67 (2014).
[Crossref]

Hu, W.

S. Hu, H. Chen, and W. Hu, “A variational solution to solitons in parity-time symmetric optical lattices,” Opt. Commun. 320, 60–67 (2014).
[Crossref]

Kartashov, Y. V.

Y. V. Kartashov, B. A. Malomed, Y. Shnir, and L. Torner, “Twisted toroidal vortex solitons in inhomogeneous media with repulsive nonlinearity,” Phys. review letters 113, 264101 (2014).
[Crossref]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton shape and mobility control in optical lattices,” Prog. Opt. 52, 63–148 (2009).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L. C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in bessel optical lattices,” Phys. Rev. Lett. 95, 10703–10710 (2005).
[Crossref]

Kivshar, Y. S.

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

Konotop, V. V.

V. M. Pérez-García, V. V. Konotop, and J. J. García-Ripoll, “Dynamics of quasicollapse in nonlinear schrödinger systems with nonlocal interactions,” Phys. Rev. E 62, 4300 (2000).
[Crossref]

Kruskal, M. D.

C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the korteweg-devries equation,” Phys. Rev. Lett. 19, 1095–1097 (1967).
[Crossref]

Lan, S.

V. Trofimov, T. Lysak, O. Matusevich, and S. Lan, “Parameter control of optical soliton in one-dimensional photonic crystal,” Math. Model. Analysis 15, 517–532 (2010).
[Crossref]

Lederer, F.

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L. C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in bessel optical lattices,” Phys. Rev. Lett. 95, 10703–10710 (2005).
[Crossref]

Li, X.

Z. Dai, Z. Yang, X. Ling, S. Zhang, Z. Pang, and X. Li, “Tripole-mode and quadrupole-mode solitons in media with a spatial exponential-decay nonlocality,” Sci. Rep. 7, 1–13 (2017).

Ling, X.

Z. Dai, Z. Yang, X. Ling, S. Zhang, Z. Pang, and X. Li, “Tripole-mode and quadrupole-mode solitons in media with a spatial exponential-decay nonlocality,” Sci. Rep. 7, 1–13 (2017).

Lisak, M.

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana 57, 917–936 (2001).
[Crossref]

K. Dimitrevski, E. Reimhult, E. Svensson, A. Öhgren, D. Anderson, A. Berntson, M. Lisak, and M. L. Quiroga-Teixeiro, “Analysis of stable self-trapping of laser beams in cubic-quintic nonlinear media,” Phys. Lett. A 248, 369–376 (1998).
[Crossref]

D. Anderson, M. Lisak, and T. Reichel, “Asymptotic propagation properties of pulses in a soliton-based optical-fiber communication system,” JOSA B 5, 207–210 (1988).
[Crossref]

A. Bondeson, M. Lisak, and D. Anderson, “Soliton perturbations: a variational principle for the soliton parameters,” Phys. Scripta 20, 479 (1979).
[Crossref]

Lopez-Aguayo, S.

Lysak, T.

V. Trofimov, T. Lysak, O. Matusevich, and S. Lan, “Parameter control of optical soliton in one-dimensional photonic crystal,” Math. Model. Analysis 15, 517–532 (2010).
[Crossref]

Malomed, B. A.

Y. V. Kartashov, B. A. Malomed, Y. Shnir, and L. Torner, “Twisted toroidal vortex solitons in inhomogeneous media with repulsive nonlinearity,” Phys. review letters 113, 264101 (2014).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L. C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in bessel optical lattices,” Phys. Rev. Lett. 95, 10703–10710 (2005).
[Crossref]

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B: Quantum Semiclassical Opt. 7, R53 (2005).
[Crossref]

B. A. Malomed, “Variational methods in nonlinear fiber optics and related fields,” Prog. optics 43, 71–194 (2002).
[Crossref]

Matusevich, O.

V. Trofimov, T. Lysak, O. Matusevich, and S. Lan, “Parameter control of optical soliton in one-dimensional photonic crystal,” Math. Model. Analysis 15, 517–532 (2010).
[Crossref]

Matusevich, O. V.

O. V. Matusevich and V. a. Trofimov, “A numerical method for calculating solitons of the nonlinear Schödinger equation in the axially symmetric case,” Mosc. Univ. Comput. Math. Cybern. 33, 117–126 (2009).
[Crossref]

O. V. Matusevich and V. A. Trofimov, “Iterative method for finding the eigenfunctions of a system of two Schrödinger equations with combined nonlinearity,” Comput. Math. Math. Phys. 48, 677–687 (2008).
[Crossref]

Mazilu, D.

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L. C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in bessel optical lattices,” Phys. Rev. Lett. 95, 10703–10710 (2005).
[Crossref]

Mihalache, D.

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L. C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in bessel optical lattices,” Phys. Rev. Lett. 95, 10703–10710 (2005).
[Crossref]

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B: Quantum Semiclassical Opt. 7, R53 (2005).
[Crossref]

Miura, R. M.

C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the korteweg-devries equation,” Phys. Rev. Lett. 19, 1095–1097 (1967).
[Crossref]

Öhgren, A.

K. Dimitrevski, E. Reimhult, E. Svensson, A. Öhgren, D. Anderson, A. Berntson, M. Lisak, and M. L. Quiroga-Teixeiro, “Analysis of stable self-trapping of laser beams in cubic-quintic nonlinear media,” Phys. Lett. A 248, 369–376 (1998).
[Crossref]

Pang, Z.

Z. Dai, Z. Yang, X. Ling, S. Zhang, Z. Pang, and X. Li, “Tripole-mode and quadrupole-mode solitons in media with a spatial exponential-decay nonlocality,” Sci. Rep. 7, 1–13 (2017).

Perez-Garcia, V.

T. Busch, J. Cirac, V. Perez-Garcia, and P. Zoller, “Stability and collective excitations of a two-component bose-einstein condensed gas: A moment approach,” Phys. Rev. A 56, 2978 (1997).
[Crossref]

Pérez-García, V. M.

V. M. Pérez-García, V. V. Konotop, and J. J. García-Ripoll, “Dynamics of quasicollapse in nonlinear schrödinger systems with nonlocal interactions,” Phys. Rev. E 62, 4300 (2000).
[Crossref]

Quiroga-Teixeiro, M. L.

K. Dimitrevski, E. Reimhult, E. Svensson, A. Öhgren, D. Anderson, A. Berntson, M. Lisak, and M. L. Quiroga-Teixeiro, “Analysis of stable self-trapping of laser beams in cubic-quintic nonlinear media,” Phys. Lett. A 248, 369–376 (1998).
[Crossref]

Reichel, T.

D. Anderson, M. Lisak, and T. Reichel, “Asymptotic propagation properties of pulses in a soliton-based optical-fiber communication system,” JOSA B 5, 207–210 (1988).
[Crossref]

Reimhult, E.

K. Dimitrevski, E. Reimhult, E. Svensson, A. Öhgren, D. Anderson, A. Berntson, M. Lisak, and M. L. Quiroga-Teixeiro, “Analysis of stable self-trapping of laser beams in cubic-quintic nonlinear media,” Phys. Lett. A 248, 369–376 (1998).
[Crossref]

Roy, S.

A. Sahoo, S. Roy, and G. P. Agrawal, “Perturbed dissipative solitons: A variational approach,” Phys. Rev. A 96, 1–8 (2017).
[Crossref]

Sahoo, A.

A. Sahoo, S. Roy, and G. P. Agrawal, “Perturbed dissipative solitons: A variational approach,” Phys. Rev. A 96, 1–8 (2017).
[Crossref]

Segev, M.

Z. Chen, M. Segev, and D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Reports on Prog. Phys. 75, 086401 (2012).
[Crossref]

Shnir, Y.

Y. V. Kartashov, B. A. Malomed, Y. Shnir, and L. Torner, “Twisted toroidal vortex solitons in inhomogeneous media with repulsive nonlinearity,” Phys. review letters 113, 264101 (2014).
[Crossref]

Skarka, V.

V. Skarka and N. Aleksić, “Stability criterion for dissipative soliton solutions of the one-, two-, and three-dimensional complex cubic-quintic ginzburg-landau equations,” Phys. review letters 96, 013903 (2006).
[Crossref]

Succi, S.

M. Chiofalo, S. Succi, and M. Tosi, “Ground state of trapped interacting bose-einstein condensates by an explicit imaginary-time algorithm,” Phys. Rev. E 62, 7438 (2000).
[Crossref]

Sukhorukov, A. A.

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

Svensson, E.

K. Dimitrevski, E. Reimhult, E. Svensson, A. Öhgren, D. Anderson, A. Berntson, M. Lisak, and M. L. Quiroga-Teixeiro, “Analysis of stable self-trapping of laser beams in cubic-quintic nonlinear media,” Phys. Lett. A 248, 369–376 (1998).
[Crossref]

Torner, L.

Y. V. Kartashov, B. A. Malomed, Y. Shnir, and L. Torner, “Twisted toroidal vortex solitons in inhomogeneous media with repulsive nonlinearity,” Phys. review letters 113, 264101 (2014).
[Crossref]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton shape and mobility control in optical lattices,” Prog. Opt. 52, 63–148 (2009).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L. C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in bessel optical lattices,” Phys. Rev. Lett. 95, 10703–10710 (2005).
[Crossref]

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B: Quantum Semiclassical Opt. 7, R53 (2005).
[Crossref]

Tosi, M.

M. Chiofalo, S. Succi, and M. Tosi, “Ground state of trapped interacting bose-einstein condensates by an explicit imaginary-time algorithm,” Phys. Rev. E 62, 7438 (2000).
[Crossref]

Trofimov, V.

V. Trofimov, T. Lysak, O. Matusevich, and S. Lan, “Parameter control of optical soliton in one-dimensional photonic crystal,” Math. Model. Analysis 15, 517–532 (2010).
[Crossref]

Trofimov, V. a.

O. V. Matusevich and V. a. Trofimov, “A numerical method for calculating solitons of the nonlinear Schödinger equation in the axially symmetric case,” Mosc. Univ. Comput. Math. Cybern. 33, 117–126 (2009).
[Crossref]

O. V. Matusevich and V. A. Trofimov, “Iterative method for finding the eigenfunctions of a system of two Schrödinger equations with combined nonlinearity,” Comput. Math. Math. Phys. 48, 677–687 (2008).
[Crossref]

Vysloukh, V. A.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton shape and mobility control in optical lattices,” Prog. Opt. 52, 63–148 (2009).
[Crossref]

Wahlquist, H. D.

H. D. Wahlquist and F. B. Estabrook, “Bäcklund transformation for solutions of the korteweg-de vries equation,” Phys. Rev. Lett. 31, 1386–1390 (1973).
[Crossref]

Wise, F.

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B: Quantum Semiclassical Opt. 7, R53 (2005).
[Crossref]

Yang, J.

J. Yang, “Newton-conjugate-gradient methods for solitary wave computations,” J. Comput. Phys. 228, 7007–7024 (2009).
[Crossref]

Yang, Z.

Z. Dai, Z. Yang, X. Ling, S. Zhang, Z. Pang, and X. Li, “Tripole-mode and quadrupole-mode solitons in media with a spatial exponential-decay nonlocality,” Sci. Rep. 7, 1–13 (2017).

Zhang, S.

Z. Dai, Z. Yang, X. Ling, S. Zhang, Z. Pang, and X. Li, “Tripole-mode and quadrupole-mode solitons in media with a spatial exponential-decay nonlocality,” Sci. Rep. 7, 1–13 (2017).

Zoller, P.

T. Busch, J. Cirac, V. Perez-Garcia, and P. Zoller, “Stability and collective excitations of a two-component bose-einstein condensed gas: A moment approach,” Phys. Rev. A 56, 2978 (1997).
[Crossref]

Comput. Math. Math. Phys. (1)

O. V. Matusevich and V. A. Trofimov, “Iterative method for finding the eigenfunctions of a system of two Schrödinger equations with combined nonlinearity,” Comput. Math. Math. Phys. 48, 677–687 (2008).
[Crossref]

J. Comput. Phys. (1)

J. Yang, “Newton-conjugate-gradient methods for solitary wave computations,” J. Comput. Phys. 228, 7007–7024 (2009).
[Crossref]

J. Opt. B: Quantum Semiclassical Opt. (1)

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B: Quantum Semiclassical Opt. 7, R53 (2005).
[Crossref]

JOSA B (1)

D. Anderson, M. Lisak, and T. Reichel, “Asymptotic propagation properties of pulses in a soliton-based optical-fiber communication system,” JOSA B 5, 207–210 (1988).
[Crossref]

Math. Model. Analysis (1)

V. Trofimov, T. Lysak, O. Matusevich, and S. Lan, “Parameter control of optical soliton in one-dimensional photonic crystal,” Math. Model. Analysis 15, 517–532 (2010).
[Crossref]

Mosc. Univ. Comput. Math. Cybern. (1)

O. V. Matusevich and V. a. Trofimov, “A numerical method for calculating solitons of the nonlinear Schödinger equation in the axially symmetric case,” Mosc. Univ. Comput. Math. Cybern. 33, 117–126 (2009).
[Crossref]

Opt. Commun. (1)

S. Hu, H. Chen, and W. Hu, “A variational solution to solitons in parity-time symmetric optical lattices,” Opt. Commun. 320, 60–67 (2014).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Lett. A (1)

K. Dimitrevski, E. Reimhult, E. Svensson, A. Öhgren, D. Anderson, A. Berntson, M. Lisak, and M. L. Quiroga-Teixeiro, “Analysis of stable self-trapping of laser beams in cubic-quintic nonlinear media,” Phys. Lett. A 248, 369–376 (1998).
[Crossref]

Phys. Rev. A (3)

A. Sahoo, S. Roy, and G. P. Agrawal, “Perturbed dissipative solitons: A variational approach,” Phys. Rev. A 96, 1–8 (2017).
[Crossref]

T. Busch, J. Cirac, V. Perez-Garcia, and P. Zoller, “Stability and collective excitations of a two-component bose-einstein condensed gas: A moment approach,” Phys. Rev. A 56, 2978 (1997).
[Crossref]

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[Crossref]

Phys. Rev. E (2)

M. Chiofalo, S. Succi, and M. Tosi, “Ground state of trapped interacting bose-einstein condensates by an explicit imaginary-time algorithm,” Phys. Rev. E 62, 7438 (2000).
[Crossref]

V. M. Pérez-García, V. V. Konotop, and J. J. García-Ripoll, “Dynamics of quasicollapse in nonlinear schrödinger systems with nonlocal interactions,” Phys. Rev. E 62, 4300 (2000).
[Crossref]

Phys. Rev. Lett. (4)

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

C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the korteweg-devries equation,” Phys. Rev. Lett. 19, 1095–1097 (1967).
[Crossref]

H. D. Wahlquist and F. B. Estabrook, “Bäcklund transformation for solutions of the korteweg-de vries equation,” Phys. Rev. Lett. 31, 1386–1390 (1973).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L. C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in bessel optical lattices,” Phys. Rev. Lett. 95, 10703–10710 (2005).
[Crossref]

Phys. review letters (2)

Y. V. Kartashov, B. A. Malomed, Y. Shnir, and L. Torner, “Twisted toroidal vortex solitons in inhomogeneous media with repulsive nonlinearity,” Phys. review letters 113, 264101 (2014).
[Crossref]

V. Skarka and N. Aleksić, “Stability criterion for dissipative soliton solutions of the one-, two-, and three-dimensional complex cubic-quintic ginzburg-landau equations,” Phys. review letters 96, 013903 (2006).
[Crossref]

Phys. Scripta (1)

A. Bondeson, M. Lisak, and D. Anderson, “Soliton perturbations: a variational principle for the soliton parameters,” Phys. Scripta 20, 479 (1979).
[Crossref]

Pramana (1)

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana 57, 917–936 (2001).
[Crossref]

Prog. Opt. (1)

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton shape and mobility control in optical lattices,” Prog. Opt. 52, 63–148 (2009).
[Crossref]

Prog. optics (1)

B. A. Malomed, “Variational methods in nonlinear fiber optics and related fields,” Prog. optics 43, 71–194 (2002).
[Crossref]

Reports on Prog. Phys. (1)

Z. Chen, M. Segev, and D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Reports on Prog. Phys. 75, 086401 (2012).
[Crossref]

Sci. Rep. (1)

Z. Dai, Z. Yang, X. Ling, S. Zhang, Z. Pang, and X. Li, “Tripole-mode and quadrupole-mode solitons in media with a spatial exponential-decay nonlocality,” Sci. Rep. 7, 1–13 (2017).

SIAM J. on Sci. Comput. (1)

W. Bao and Q. Du, “Computing the ground state solution of bose–einstein condensates by a normalized gradient flow,” SIAM J. on Sci. Comput. 25, 1674–1697 (2004).
[Crossref]

Other (1)

Y. S. Kivshar, G. P. Agrawal, and G. P. Agrawal, eds., Index (Academic Press, Burlington, 2003).

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Figures (4)

Fig. 1
Fig. 1 The amplitude (a) and beam’s width (b), predicted by the numerical variational procedure for the fundamental soliton (m = 0), compared to the results of the analytical variational approximation (see the text). The maximum relative differences for parameters A1 and A2 are eA1 = 1.72 × 10−8 % and eA2 = 2.40 × 10−8 %. The same comparison for the amplitude (c) and width (d) of the vortex beam with m = 1 in the Kerr medium, with maximum relative differences eA1 = 2.17 × 10−7 % and eA2 = 1.65 × 10−7 %. Comparison between the exact soliton shape and the variationally obtained using Eq. (7) for fundamental (e) and single vortex soliton (f), in both cases λ = 0.5.
Fig. 2
Fig. 2 Simulated propagation in the medium with the combined nonlinearity corresponding to Eq. (14) + Eq. (16), initiated by inputs produced by the numerically implemented variational approximation. (a) The vortex soliton with topological charge m = 1 with A1,2,3 = (0.7830, 2.7903, 2.7903) and TDA 15 × 15. (b) The propagation of the quadrupole soliton in the medium with the combined nonlinearity corresponding to Eq. (14) + Eq. (16), with A1,2,3 = (0.5767, 3.4560, 3.4560) and TDA 20 × 20. The peak-intensity evolution is shown in the right-hand column.
Fig. 3
Fig. 3 The same as in Fig. 2, but under the action of the combined nonlinearity given by Eq. (15) + Eq. (16). (a) The vortex with m = 1, A1,2,3 = (0.7860, 2.7915, 2.7915), and TDA 15 × 15. (b) The azimuthon with m = 2, δ = π/3, A1,2,3 = (0.63574, 3.4760, 3.4760), and TDA 20 × 20. (c) The asymmetric quadrupole with s = 0.2, λ = 0.8, = 5π/16, A1,2,3 = (1.0334, 3.4235, 2.4701), and TDA 20 × 20.
Fig. 4
Fig. 4 The same as in Fig. 2, but in the medium with the combination of the nonlinear terms corresponding to the combination of Eqs. (15) + (16), with s = 0.05, p = 15 and k = 1. (a) The fourth-order azimuthon with δ = π/3, A1,2,3 = (0.4216, 2.8725, 2.8725), and TDA 20 × 20. (b) The asymmetric quadrupole with = 5π/16, A1,2,3 = (1.2722, 2.2057, 1.3310), and TDA 15 × 15.

Tables (1)

Tables Icon

Table 1 Anharmonic terms in the Lagrangian

Equations (16)

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S = d t d r = L d t ,
d d t ( Ψ t * ) + r i = x , y , z d d r i ( Ψ r i * ) Ψ * = 0 ,
i t Ψ + 2 Ψ + N ( | Ψ | 2 , r ) Ψ = 0 ,
= i 2 ( Ψ t Ψ * Ψ * t Ψ ) + | Ψ | 2 + 𝒩 ( | Ψ | 2 , r ) ,
d d t ( L ( A ) ( d A n / d t ) ) L ( A ) A n = 0 ,
H L ( A ) = [ A 1 2 L ( A ) A 2 A 1 L ( A ) A n A 1 L ( A ) A 1 A 2 L ( A ) A 2 2 L ( A ) A n A 2 L ( A ) A 1 A n L ( A ) A 2 A n L ( A ) A n 2 L ( A ) ] .
A ( i + 1 ) = A ( i ) H L ( A ) 1 A L ( A ) ,
L ( A ) A n = ( A ) A n d r ( A + Δ A n ) ( A Δ A n ) 2 Δ A n d r = L ( A + Δ A n ) L ( A Δ A n ) 2 Δ A n ,
2 L ( A ) A n A m L ( A + Δ A n / 2 + Δ A m / 2 ) L ( A Δ A n / 2 + Δ A m / 2 ) L ( A + Δ A n / 2 Δ A m / 2 ) + L ( A Δ A n / 2 Δ A m / 2 ) Δ A n Δ A m .
Ψ ( x , y , z ) = ψ ( r ) e i λ z ,
λ ψ ( r ) + 2 ψ ( r ) + N ( | ψ | 2 , r ) ψ ( r ) = 0 ,
= λ | ψ | 2 + | ψ | 2 + 𝒩 ( | ψ | 2 , r ) .
𝒩 ( | ψ | 2 , r ) = 1 / 2 | ψ | 4 ,
𝒩 ( | ψ | 2 , r ) = ( ln ( s | ψ | 2 + 1 ) s | ψ | 2 ) / s 2 ,
𝒩 ( | ψ | 2 , r ) = p [ J n ( k x 2 + y 2 ) ] 2 | ψ | 2 .
ψ ( r ) = A 1 ( m + 1 ) / 2 exp [ ( x / A 2 ) 2 ( y / A 3 ) 2 ] × [ ( cos 2 ) x 2 + ( sin 2 ) y 2 ] m / 2 [ ( cos δ ) cos ( m θ ) + i ( sin δ ) sin ( m θ ) ] ,

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