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On the Properties of a Nonlocal Nonlinear Schrödinger Model and Its Soliton Solutions

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Applications of Nonlinear Analysis

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 134))

Abstract

Nonlinear waves are normally described by means of certain nonlinear evolution equations. However, finding physically relevant exact solutions of these equations is, in general, particularly difficult. One of the most known nonlinear evolution equation is the nonlinear Schrödinger (NLS), a universal equation appearing in optics, Bose-Einstein condensates, water waves, plasmas, and many other disciplines. In optics, the NLS system is used to model a unique balance between the critical effects that govern propagation in dispersive nonlinear media, namely dispersion/diffraction and nonlinearity. This balance leads to the formation of solitons, namely robust localized waveforms that maintain their shape even when they interact. However, for several physically relevant contexts the standard NLS equation turns out to be an oversimplified description. This occurs in the case of nonlocal media, such as nematic liquid crystals, plasmas, and optical media exhibiting thermal nonlinearities. Here, we study the properties and soliton solutions of such a nonlocal NLS system, composed by a paraxial wave equation for the electric field envelope and a diffusion-type equation for the medium’s refractive index. The study of this problem is particularly interesting since remarkable properties of the traditional NLS—associated with complete integrability—are lost in the nonlocal case. Nevertheless, we identify cases where derivation of exact solutions is possible while, in other cases, we resort to multiscale expansions methods. The latter, allows us to reduce this systems to a known integrable equation with known solutions, which in turn, can be used to approximate the solutions of the initial system. By doing so, a plethora of solutions can be found; solitary waves solutions, planar or ring-shaped, and of dark or anti-dark type, are predicted to occur.

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References

  1. M.J. Ablowitz, Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons (Cambridge University Press, Cambridge, 2011)

    Book  Google Scholar 

  2. M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge University Press, Cambridge, 1991)

    Book  Google Scholar 

  3. M.J. Ablowitz, T.P. Horikis, Interacting nonlinear wave envelopes and rogue wave formation in deep water. Phys. Fluids 27, 012107 (2015)

    Article  Google Scholar 

  4. G.P. Agrawal, Nonlinear Fiber Optics (Academic, Cambridge, 2013)

    MATH  Google Scholar 

  5. M. Aguero, D.J. Frantzeskakis, P.G. Kevrekidis, Asymptotic reductions of two coupled (2+1)-dimensional nonlinear Schrödinger equations: application to Bose-Einstein condensates. J. Phys. A Math. Gen. 39, 7705–7718 (2006)

    Article  Google Scholar 

  6. A. Alberucci, G. Assanto, Modeling nematicon propagation. Mol. Cryst. Liq. Cryst. 572, 2–12 (2013)

    Article  Google Scholar 

  7. A. Alberucci, M. Peccianti, G. Assanto, A. Dyadyusha, M. Kaczmarek, Two-color vector solitons in nonlocal media. Phys. Rev. Lett. 97, 153903 (2006)

    Article  Google Scholar 

  8. A. Armaroli, S. Trillo, Suppression of transverse instabilities of dark solitons and their dispersive shock waves. Phys. Rev. A 80, 053803 (2009)

    Article  Google Scholar 

  9. G. Assanto, Nematicons: Spatial Optical Solitons in Nematic Liquid Crystals (Wiley, Hoboken, 2012)

    Book  Google Scholar 

  10. G. Assanto, A.A. Minzoni, N.F. Smyth, Light self-localization in nematic liquid crystals: modelling solitons in nonlocal reorientational media. J. Nonlinear Opt. Phys. Mater. 18, 657–691 (2009)

    Article  Google Scholar 

  11. G. Assanto, N.F. Smyth, A.L. Worthy, Two-color, nonlocal vector solitary waves with angular momentum in nematic liquid crystals. Phys. Rev. A 78, 013832 (2008)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. A.V. Buryak, P. Di Trapani, D.V. Skryabin, S. Trillo, Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications. Phys. Rep. 370, 63–235 (2002)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  15. E.G. Charalampidis, P.G. Kevrekidis, D.J. Frantzeskakis, B.A. Malomed, Dark-bright solitons in coupled nonlinear Schrödinger equations with unequal dispersion coefficients. Phys. Rev. E 91, 012924 (2015)

    Article  Google Scholar 

  16. C. Conti, M. Peccianti, G. Assanto, Route to nonlocality and observation of accessible solitons. Phys. Rev. Lett. 91, 073901 (2003)

    Article  Google Scholar 

  17. T. Dauxois, M. Peyrard, Physics of Solitons (Cambridge University Press, Cambridge, 2006)

    MATH  Google Scholar 

  18. A. Dreischuh, D.N. Neshev, G.G. Paulus, F. Grasbon, H. Walther, Ring dark solitary waves: experiment versus theory. Phys. Rev. E 66, 066611 (2002)

    Article  Google Scholar 

  19. A. Dreischuh, D.N. Neshev, D.E. Petersen, O. Bang, W. Krolikowski, Observation of attraction between dark solitons. Phys. Rev. Lett. 96, 043901 (2006)

    Article  Google Scholar 

  20. G. El, N.F. Smyth, Radiating dispersive shock waves in non-local optical media. Proc. Roy. Soc. Lond. A 472, 20150633 (2016)

    Article  MathSciNet  Google Scholar 

  21. D.J. Frantzeskakis, Vector solitons supported by the third-order dispersion. Phys. Lett. A 285, 363–367 (2001)

    Article  Google Scholar 

  22. D.J. Frantzeskakis, Dark solitons in Bose-Einstein condensates: from theory to experiments. J. Phys. A Math. Theor. 43, 213001 (2010)

    Article  MathSciNet  Google Scholar 

  23. D.J. Frantzeskakis, B.A. Malomed, Multiscale expansions for a generalized cylindrical nonlinear Schrödinger equation. Phys. Lett. A 264, 179–185 (1999)

    Article  MathSciNet  Google Scholar 

  24. A. Hasegawa, Y. Kodama, Solitons in Optical Communications (Claredon Press, Oxford, 1995)

    MATH  Google Scholar 

  25. R. Hirota, Exact solutions to the equation describing “cylindrical solitons”. Phys. Lett. A 71, 393–394 (1979)

    Article  MathSciNet  Google Scholar 

  26. T.P. Horikis, Small-amplitude defocusing nematicons. J. Phys. A Math. Theor. 48, 02FT01 (2015)

    Google Scholar 

  27. T.P. Horikis, D.J. Frantzeskakis, On the NLS to KdV connection. Rom. J. Phys. 59, 195–203 (2014)

    Google Scholar 

  28. T.P. Horikis, D.J. Frantzeskakis, Asymptotic reductions and solitons of nonlocal nonlinear Schrödinger equations. J. Phys. A Math. Theor. 49, 205202 (2016)

    Article  Google Scholar 

  29. E. Infeld, G. Rowlands, Nonlinear Waves, Solitons and Chaos (Cambridge University Press, Cambridge, 1990)

    MATH  Google Scholar 

  30. A. Jeffrey, T. Kawahara, Asymptotic Methods in Nonlinear Wave Theory (Pitman Books, London, 1982)

    MATH  Google Scholar 

  31. R.S. Johnson, Water waves and Korteweg-de Vries equations. J. Fluid Mech. 97, 701–719 (1980)

    Article  MathSciNet  Google Scholar 

  32. R.S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves (Cambridge University Press, Cambridge, 1997)

    Book  Google Scholar 

  33. R.S. Johnson, A note on an asymptotic solution of the cylindrical Korteweg-de Vries equation. Wave Motion 30, 1–16 (1999)

    Article  MathSciNet  Google Scholar 

  34. Y.N. Karamzin, A.P. Sukhorukov, Nonlinear interaction of diffracted light beams in a medium with quadratic nonlinearity: mutual focusing of beams and limitation on the efficiency of optical frequency converters. JETP Lett. 20, 339–342 (1974)

    Google Scholar 

  35. V.I. Karpman, Non-linear Waves in Dispersive Media (Elsevier, Amsterdam, 1974)

    Google Scholar 

  36. Y.V. Kartashov, L. Torner, Gray spatial solitons in nonlocal nonlinear media. Opt. Lett. 32, 946–948 (2007)

    Article  Google Scholar 

  37. P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-González, The Defocusing Nonlinear Schrödinger Equation: From Dark Solitons to Vortices and Vortex Rings (SIAM, Philadelphia, 2015)

    Book  Google Scholar 

  38. K.R. Khusnutdinova, C. Klein, V.B. Matveev, A.O. Smirnov, On the integrable elliptic cylindrical Kadomtsev-Petviashvili equation. Chaos 23, 013126 (2013)

    Article  MathSciNet  Google Scholar 

  39. Y.S. Kivshar, G.P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, Cambridge, 2003)

    Google Scholar 

  40. Y.S. Kivshar, B. Luther-Davies, Dark optical solitons: physics and applications. Phys. Rep. 298, 81–197 (1998)

    Article  Google Scholar 

  41. Y.S. Kivshar, D.E. Pelinovsky, Self-focusing and transverse instabilities of solitary waves. Phys. Rep. 331, 117–195 (2000)

    Article  MathSciNet  Google Scholar 

  42. Y.S. Kivshar, X. Yang, Ring dark solitons. Phys. Rev. E 50, R40–R43 (1994)

    Article  Google Scholar 

  43. K. Ko, H.H. Kuehl, Cylindrical and spherical Korteweg-deVries solitary waves. Phys. Fluids 22, 1343–1348 (1979)

    Article  MathSciNet  Google Scholar 

  44. W. Krolikowski, O. Bang, J.J. Rasmussen, J. Wyller, Modulational instability in nonlocal nonlinear kerr media. Phys. Rev. E 64, 016612 (2001)

    Article  Google Scholar 

  45. W. Krolikowski, O. Bang, N.I. Nikolov, D.N. Neshev, J. Wyller, J.J. Rasmussen, D. Edmundson, Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media. J. Opt. B Quantum Semiclassical Opt. 6, S288–S294 (2004)

    Article  Google Scholar 

  46. E.A. Kuznetsov, S.K. Turitsyn, Two- and three-dimensional solitons in weakly dispersive media. J. Exp. Theor. Phys. 55, 844–847 (1982)

    Google Scholar 

  47. E.A. Kuznetsov, S.K. Turitsyn, Instability and collapse of solitons in media with a defocusing nonlinearity. J. Exp. Theor. Phys. 67, 1583–1588 (1988)

    MathSciNet  Google Scholar 

  48. A.G. Litvak, V.A. Mironov, G.M. Fraiman, A.D. Yunakovskii, Thermal self-effect of wave beams in a plasma with a nonlocal nonlinearity. Sov. J. Plasma Phys. 1, 60–71 (1975)

    Google Scholar 

  49. J.M.L. MacNeil, N.F. Smyth, G. Assanto, Exact and approximate solutions for optical solitary waves in nematic liquid crystals. Phys. D 284, 1–15 (2014)

    Article  MathSciNet  Google Scholar 

  50. B.A. Malomed, V.I. Shrira, Soliton caustics. Phys. D 53, 1–12 (1991)

    Article  MathSciNet  Google Scholar 

  51. V.K. Mel’nikov, On equations for wave interactions. Lett. Math. Phys. 7, 129–136 (1983)

    Article  MathSciNet  Google Scholar 

  52. V.K. Mel’nikov, Exact solutions of the korteweg-de vries equation with a self-consistent source. Phys. Lett. A 128, 488–492 (1988)

    Article  MathSciNet  Google Scholar 

  53. V.K. Mel’nikov, Integration method of the korteweg-de vries equation with a self-consistent source. Phys. Lett. A 133, 493–496 (1988)

    Article  MathSciNet  Google Scholar 

  54. D. Mihalache, Multidimensional solitons and vortices in nonlocal noninear optical media. Rom. Rep. Phys. 59, 515–522 (2007)

    Google Scholar 

  55. D. Mihalache, D. Mazilu, F. Lederer, B.A. Malomed, Y.V. Kartashov, L.-C. Crasovan, L. Torner, Three-dimensional spatiotemporal optical solitons in nonlocal nonlinear media. Phys. Rev. E 73, 025601(R) (2006)

    Google Scholar 

  56. A. Minovich, D.N. Neshev, A. Dreischuh, W. Krolikowski, Y.S. Kivshar, Experimental reconstruction of nonlocal response of thermal nonlinear optical media. Opt. Lett. 32, 1599–1601 (2007)

    Article  Google Scholar 

  57. H.E. Nistazakis, D.J. Frantzeskakis, B.A. Malomed, P.G. Kevrekidis, Head-on collisions of ring dark solitons. Phys. Lett. A 285, 157–164 (2001)

    Article  Google Scholar 

  58. W. Oevel, W.-H. Steeb, Painleve analysis for a time-dependent Kadomtsev-Petviashvili equation. Phys. Lett. A 103, 239–242 (1984)

    Article  MathSciNet  Google Scholar 

  59. M. Peccianti, G. Assanto, Nematicons. Phys. Rep. 516, 147–208 (2012)

    Article  Google Scholar 

  60. P. Pedri, L. Santos, Two-dimensional bright solitons in dipolar Bose-Einstein condensates. Phys. Rev. Lett. 95, 200404 (2005)

    Article  Google Scholar 

  61. D.E. Pelinovsky, Y.A. Stepanyants, Solitary wave instability in the positive-dispersion media described by the two-dimensional Boussinesq equations. J. Exp. Theor. Phys. 79, 105–112 (1993)

    Google Scholar 

  62. D.E. Pelinovsky, Y.A. Stepanyants, Y.S. Kivshar, Self-focusing of plane dark solitons in nonlinear defocusing media. Phys. Rev. E 51, 5016–5026 (1995)

    Article  MathSciNet  Google Scholar 

  63. A. Piccardi, A. Alberucci, N. Tabiryan, G. Assanto, Dark nematicons. Opt. Lett. 36, 1356–1358 (2011)

    Article  Google Scholar 

  64. M. Remoissenet, Waves Called Solitons (Springer, Berlin, 1999)

    Book  Google Scholar 

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

    Article  Google Scholar 

  66. B.D. Skuse, N.F. Smyth, Two-color vector-soliton interactions in nematic liquid crystals in the local response regime. Phys. Rev. A 77, 013817 (2008)

    Article  Google Scholar 

  67. B.D. Skuse, N.F. Smyth, Interaction of two-color solitary waves in a liquid crystal in the nonlocal regime. Phys. Rev. A 79, 063806 (2009)

    Article  Google Scholar 

  68. J.C.F. Sturm, Mémoire sur la résolution des équations numériques. Bull. Sci. Férussac 11, 419–425 (1829)

    Google Scholar 

  69. D. Suter, T. Blasberg, Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium. Phys. Rev. A 48, 4583–4587 (1993)

    Article  Google Scholar 

  70. F. Tsitoura, V. Achilleos, B.A. Malomed, D. Yan, P.G. Kevrekidis, D.J. Frantzeskakis. Matter-wave solitons in the counterflow of two immiscible superfluids. Phys. Rev. A 87, 063624 (2013)

    Article  Google Scholar 

  71. T. Tsuzuki, Nonlinear waves in the Pitaevskii-Gross equation. J. Low Temp. Phys. 4, 441–457 (1971)

    Article  Google Scholar 

  72. S.K. Turitsyn, Spatial dispersion of nonlinearity and stability of many dimensional solitons. Theor. Math. Phys. 64, 797–801 (1985)

    Article  Google Scholar 

  73. A.I. Yakimenko, Y.A. Zaliznyak, Y.S. Kivshar, Stable vortex solitons in nonlocal self-focusing nonlinear media. Phys. Rev. E 71, 065603(R) (2005)

    Google Scholar 

  74. V.E. Zakharov, E.A. Kuznetsov, Multi-scale expansions in the theory of systems integrable by the inverse scattering transform. Phys. D 18, 455–463 (1986)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Ioannina, Ioannina, Greece

    Theodoros P. Horikis

  2. Department of Physics, National and Kapodistrian University of Athens, Athens, Greece

    Dimitrios J. Frantzeskakis

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  1. Theodoros P. Horikis
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  2. Dimitrios J. Frantzeskakis
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Correspondence to Theodoros P. Horikis .

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  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

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Horikis, T.P., Frantzeskakis, D.J. (2018). On the Properties of a Nonlocal Nonlinear Schrödinger Model and Its Soliton Solutions. In: Rassias, T. (eds) Applications of Nonlinear Analysis . Springer Optimization and Its Applications, vol 134. Springer, Cham. https://doi.org/10.1007/978-3-319-89815-5_14

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