Symmetry analysis and rogue wave solutions for the (2+1)-dimensional nonlinear Schrödinger equation with variable coefficients

Abstract This paper addresses ( 2 + 1 )-dimensional nonlinear Schrodinger equation (NLSE). For the special case, linear Schrodinger equation (LSE), it can be transformed into the same form of equation. On the basis of different gauge constraint, we construct potential symmetries for the LSE. And then, we consider ( 2 + 1 )-dimensional NLSE using Lie symmetry analysis. By means of similarity transformations, we study the ( 2 + 1 )-dimensional NLSE with nonlinearities and potentials depending on time as well as on the spatial coordinates. At last, we present the rouge wave solutions of ( 2 + 1 )-dimensional NLSE.

[1]  P. Olver.  Applications of lie groups to differential equations , 1986 .

[2]  V. Zakharov,et al.  Exact Theory of Two-dimensional Self-focusing and One-dimensional Self-modulation of Waves in Nonlinear Media , 1970 .

[3]  Randall G. Hulet,et al.  Formation and propagation of matter-wave soliton trains , 2002, Nature.

[4]  K. Fakhar,et al.  Lie symmetry analysis, nonlinear self-adjointness and conservation laws to an extended (2+1)-dimensional Zakharov–Kuznetsov–Burgers equation , 2015 .

[5]  C. Dai,et al.  Controllable mechanism of breathers in the (2+1)-dimensional nonlinear Schrödinger equation with different forms of distributed transverse diffraction , 2014 .

[6]  Phillips,et al.  Generating solitons by phase engineering of a bose-einstein condensate , 2000, Science.

[7]  Roman O. Popovych,et al.  Admissible Transformations and Normalized Classes of Nonlinear Schrödinger Equations , 2010 .

[8]  A. Kara,et al.  Nonlocal symmetry analysis and conservation laws to an third-order Burgers equation , 2016 .

[9]  G. Bluman,et al.  Nonlocal symmetries and nonlocal conservation laws of Maxwell’s equations , 1997 .

[10]  Z. Deng,et al.  Multi-symplectic method to simulate Soliton resonance of (2+1)-dimensional Boussinesq equation , 2013 .

[11]  T. Bridges.  Multi-symplectic structures and wave propagation , 1997, Mathematical Proceedings of the Cambridge Philosophical Society.

[12]  A. Kara,et al.  Symmetry analysis and conservation laws for the class of time-fractional nonlinear dispersive equation , 2015 .

[13]  Bo Wang,et al.  Chaos in an embedded single-walled carbon nanotube , 2013 .

[14]  S. Reich.  Multi-Symplectic Runge—Kutta Collocation Methods for Hamiltonian Wave Equations , 2000 .

[15]  J. Belmonte-Beitia.  Symmetric and asymmetric bound states for the nonlinear Schrödinger equation with inhomogeneous nonlinearity , 2009 .

[16]  Juan Belmonte-Beitia,et al.  Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities. , 2006, Physical review letters.

[17]  G. Bluman,et al.  Applications of Symmetry Methods to Partial Differential Equations , 2009 .

[18]  Gabriel F. Calvo,et al.  Exact solutions for the quintic nonlinear Schrödinger equation with time and space modulated nonlinearities and potentials , 2009 .

[19]  A. Kara,et al.  Nonlocal symmetry analysis, explicit solutions and conservation laws for the fourth-order Burgers' equation , 2015 .

[20]  Weipeng Hu,et al.  Competition between geometric dispersion and viscous dissipation in wave propagation of KdV-Burgers equation , 2015 .

[21]  Conservative parameterization schemes , 2012, 1209.4279.

[22]  C. Dai,et al.  Exact spatial similaritons and rogons in 2D graded-index waveguides. , 2010, Optics letters.

[23]  G. Bluman,et al.  New classes of Schrödinger equations equivalent to the free particle equation through non-local transformations , 1996 .

[24]  C'elestin Kurujyibwami.  Equivalence groupoid for (1+2)-dimensional linear Schrödinger equations with complex potentials , 2015 .

[25]  G. Bluman,et al.  Symmetries and differential equations , 1989 .

[26]  G. Bluman,et al.  Multidimensional partial differential equation systems: Generating new systems via conservation laws, potentials, gauges, subsystems , 2010 .

[27]  Roman O. Popovych,et al.  Invariant Discretization Schemes for the Shallow-Water Equations , 2012, SIAM J. Sci. Comput..

[28]  B Eiermann,et al.  Bright Bose-Einstein gap solitons of atoms with repulsive interaction. , 2004, Physical review letters.

[29]  Stegeman,et al.  Optical Spatial Solitons and Their Interactions: Universality and Diversity. , 1999, Science.

[30]  J. Belmonte-Beitia.  Exact solutions for the quintic nonlinear Schrödinger equation with inhomogeneous nonlinearity , 2009 .

[31]  Yu Zhang,et al.  Multi-symplectic method for peakon-antipeakon collision of quasi-Degasperis-Procesi equation , 2014, Comput. Phys. Commun..

[32]  Songmei Han,et al.  Generalized multi-symplectic integrators for a class of Hamiltonian nonlinear wave PDEs , 2013, J. Comput. Phys..

[33]  Weipeng Hu,et al.  Chaos in embedded fluid-conveying single-walled carbon nanotube under transverse harmonic load series , 2015 .