The European Physical Journal D

, Volume 58, Issue 1, pp 141–146 | Cite as

Formation of fundamental solitons in the two-dimensional nonlinear Schrödinger equation with a lattice potential

Nonlinear Dynamics

Abstract

We consider self-trapping of 2D solitons in the model based on the Gross-Pitaevskii/nonlinear Schrödinger equation with the self-attractive cubic nonlinearity and a periodic potential of the optical-lattice (OL) type. It is known that this model may suppress the collapse, giving rise to a family of stable fundamental solitons. Here, we report essential dynamical features of self-trapping of the fundamental solitons from input configurations of two types, with vorticity 0 or 1. We identify regions in the respective parameter spaces corresponding to the formation of the soliton, collapse, and decay. A noteworthy result is the self-trapping of stable fundamental solitons in cases when the input norm essentially exceeds the collapse threshold. We also compare predictions of the dynamical variational approximation with direct numerical simulations.

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References

  1. 1.
    B.A. Malomed, D. Mihalache, F. Wise, L. Torner, J. Opt. B: Quantum Semiclass. Opt. 7, R53 (2005)CrossRefADSGoogle Scholar
  2. 2.
    B.B. Baizakov, V.V. Konotop, M. Salerno, J. Phys. B: At. Mol. Opt. Phys. 35, 5105 (2002)CrossRefADSGoogle Scholar
  3. 3.
    E.A. Ostrovskaya, Yu.S. Kivshar, Phys. Rev. Lett. 90, 160407 (2003)CrossRefADSGoogle Scholar
  4. 4.
    H. Sakaguchi, B.A. Malomed, J. Phys. B 37, 2225 (2004)CrossRefADSGoogle Scholar
  5. 5.
    Z. Shi, J. Wang, Z. Chen, J. Yang, Phys. Rev. A 78, 063812 (2008)CrossRefADSGoogle Scholar
  6. 6.
    J.W. Fleischer, M. Segev, N.K. Efremidis, D.N. Christodoulides, Nature 422, 147 (2003)CrossRefADSGoogle Scholar
  7. 7.
    D.N. Neshev, T.J. Alexander, E.A. Ostrovskaya, Y.S. Kivshar, H. Martin, I. Makasyuk, Z. Chen, Phys. Rev. Lett. 92, 123903 (2004)CrossRefADSGoogle Scholar
  8. 8.
    J.W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, D.N. Christodoulides, Phys. Rev. Lett. 92, 123904 (2004)CrossRefADSGoogle Scholar
  9. 9.
    P. Xie, Z.-Q. Zhang, X. Zhang, Phys. Rev. E 67, 026607 (2003)CrossRefADSGoogle Scholar
  10. 10.
    A. Ferrando, M. Zacarés, P.F. de Cordoba, D. Binosi, J. A. Monsoriu, Opt. Express 11, 452 (2003)CrossRefADSGoogle Scholar
  11. 11.
    A. Ferrando, M. Zacarés, P.F. de Cordoba, D. Binosi, J.A. Monsoriu, Opt. Express 12, 817 (2004)CrossRefADSGoogle Scholar
  12. 12.
    B.B. Baizakov, B.A. Malomed, M. Salerno, Europhys. Lett. 63, 642 (2003)CrossRefADSGoogle Scholar
  13. 13.
    G.D. Montesinos, V.M. Pérez-García, Math. Comput. Simul. 69, 447 (2005)MATHCrossRefGoogle Scholar
  14. 14.
    S. Blanes, P.C. Moan, J. Comput. Appl. Math. 142, 313 (2002)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    J. Yang, Z.H. Musslimani, Opt. Lett. 28, 2094 (2003)CrossRefADSGoogle Scholar
  16. 16.
    Z.H. Musslimani, J. Yang, J. Opt. Soc. Am. B 21, 973 (2004)CrossRefADSGoogle Scholar
  17. 17.
    H. Sakaguchi, B.A. Malomed, Europhys. Lett. 72, 698 (2005)CrossRefMathSciNetADSGoogle Scholar
  18. 18.
    A. Gubeskys, B.A. Malomed, Phys. Rev. A 76, 043623 (2007)CrossRefMathSciNetADSGoogle Scholar
  19. 19.
    J. Wang, J. Yang, Phys. Rev. A 77, 033834 (2008)CrossRefADSGoogle Scholar
  20. 20.
    B.B. Baizakov, B.A. Malomed, M. Salerno, Phys. Rev. A 70, 053613 (2004)CrossRefADSGoogle Scholar
  21. 21.
    T. Mayteevarunyoo, B.A. Malomed, Phys. Rev. E 73, 036615 (2006)CrossRefADSGoogle Scholar
  22. 22.
    L. Gubeskys, B.A. Malomed, Phys. Rev. A 79, 045801 (2009)CrossRefADSGoogle Scholar
  23. 23.
    T. Mayteevarunyoo, B.A. Malomed, B.B. Baizakov, M. Salerno, Physica D 238, 1439 (2009)MATHCrossRefADSGoogle Scholar
  24. 24.
    D. Mihalache, D. Mazilu, F. Lederer, Y.V. Kartashov, L.-C. Crasovan, L. Torner, Phys. Rev. E 70, 055603(R) (2004)CrossRefADSGoogle Scholar
  25. 25.
    M. Trippenbach, M. Matuszewski, B.A. Malomed, Europhys. Lett. 70, 8 (2005)CrossRefADSGoogle Scholar
  26. 26.
    M. Matuszewski, E. Infeld, B.A. Malomed, M. Trippenbach, Phys. Rev. Lett. 95, 050403 (2005)CrossRefADSGoogle Scholar
  27. 27.
    D. Mihalache, D. Mazilu, F. Lederer, B.A. Malomed, Y.V. Kartashov, L.-C. Crasovan, L. Torner, Phys. Rev. Lett. 95, 023902 (2005)CrossRefADSGoogle Scholar
  28. 28.
    H. Leblond, B.A. Malomed, D. Mihalache, Phys. Rev. E 76, 026604 (2007)CrossRefMathSciNetADSGoogle Scholar
  29. 29.
    B. Terhalle, T. Richter, J.J.H. Law, D. Gories, P. Rose, T.J. Alexander, P.G. Kevrekidis, A.S. Desyatnikov, W. Królikowski, F. Kaiser, C. Denz, Y.S. Kivshar, Phys. Rev. A 79, 043821 (2009)CrossRefADSGoogle Scholar
  30. 30.
    B.A. Malomed, Progress in Optics 43, 69 (2002)Google Scholar
  31. 31.
    G. Kalosakas, K.Ö. Rasmussen, A.R. Bishop, Phys. Rev. Lett. 89, 030402 (2002)CrossRefADSGoogle Scholar
  32. 32.
    B.B. Baizakov, M. Salerno, Phys. Rev. A 69, 013602 (2004)CrossRefADSGoogle Scholar
  33. 33.
    T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, A. Tünnermann, Opt. Lett. 29, 468 (2004)CrossRefADSGoogle Scholar
  34. 34.
    A. Szameit, D. Blömer, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, Opt. Express 13, 10552 (2005)CrossRefADSGoogle Scholar
  35. 35.
    L. Bergé, Phys. Rep. 303, 259 (1998)CrossRefMathSciNetADSGoogle Scholar
  36. 36.
    M. Desaix, D. Anderson, M. Lisak, J. Opt. Soc. Am. 8, 2082 (1991)CrossRefADSGoogle Scholar
  37. 37.
    O. Morsch, M. Oberthaler, Rev. Mod. Phys. 78, 179 (2006)CrossRefADSGoogle Scholar
  38. 38.
    K.E. Strecker, G.B. Partridge, A.G. Truscott, R.G. Hulet, Nature 417, 150 (2002)CrossRefADSGoogle Scholar
  39. 39.
    L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L.D. Carr, Y. Castin, C. Salomon, Science 296, 1290 (2002)CrossRefADSGoogle Scholar
  40. 40.
    A. Gubeskys, B.A. Malomed, Phys. Rev. A 75, 063602 (2007)CrossRefMathSciNetADSGoogle Scholar
  41. 41.
    R. Driben, B.A. Malomed, A. Gubeskys, J. Zyss, Phys. Rev. E 76, 066604 (2007)CrossRefMathSciNetADSGoogle Scholar
  42. 42.
    H. Sakaguchi, B.A. Malomed, Europhys. Lett. 72, 698 (2005)CrossRefMathSciNetADSGoogle Scholar
  43. 43.
    H. Sakaguchi, B.A. Malomed, Phys. Rev. A 79, 043606 (2009)CrossRefADSGoogle Scholar
  44. 44.
    V.I. Kruglov, V.M. Volkov, R.A. Vlasov, V.V. Drits, J. Phys. A 21, 4381 (1988)MATHCrossRefMathSciNetADSGoogle Scholar
  45. 45.
    V.I. Kruglov, Y.A. Logvin, V.M. Volkov, J. Mod. Phys. 39, 2277 (1992)MATHMathSciNetADSGoogle Scholar
  46. 46.
    L. Sanchez-Palencia, D. Clément, P. Lugan, P. Bouyer, G.V. Shlyapnikov, A. Aspect, Phys. Rev. Lett. 98, 210401 (2007)CrossRefADSGoogle Scholar
  47. 47.
    L. Fallani, J.E. Lye, V. Guarrera, C. Fort, M. Inguscio, Phys. Rev. Lett. 98, 130404 (2007)CrossRefADSGoogle Scholar
  48. 48.
    Y.P. Chen, J. Hitchcock, D. Dries, M. Junker, C. Welford, R.G. Hulet, Phys. Rev. A 77, 033632 (2008)CrossRefADSGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of MassachusettsAmherstUSA
  2. 2.Department of Physical ElectronicsSchool of Electrical Engineering, Faculty of Engineering, Tel Aviv UniversityTel AvivIsrael

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