Bright and dark soliton solutions and Bäcklund transformation for the Eckhaus-Kundu equation with the cubic-quintic nonlinearity

In this paper, with symbolic computation, the Eckhaus-Kundu equation which appears in the quantum field theory, weakly nonlinear dispersive water waves and nonlinear optics, is studied via the Hirota method. By virtue of the dependent variable transformation, the bilinear form is obtained. Bilinear Backlund transformation is given with the help of exchange formulae and the corresponding one-soliton solution is derived. Bright and dark N-soliton solutions are obtained. Propagation and interaction of the bright and dark solitons are discussed analytically and graphically. Interactions of the two solitons are presented. Bound state of the two solitons can be suppressed via the choice of parameters.

[1]  Shun-dong Zhu,et al.  Exact solutions for the high-order dispersive cubic-quintic nonlinear Schrödinger equation by the extended hyperbolic auxiliary equation method , 2007 .

[2]  Limin Wang,et al.  New soliton-like solutions for KdV equation with variable coefficient , 2005 .

[3]  R. S. Johnson.  On the modulation of water waves in the neighbourhood of kh ≈ 1.363 , 1977, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[4]  R. Hirota,et al.  Hierarchies of Coupled Soliton Equations. I , 1991 .

[5]  Dug Young Kim,et al.  Experimental observation of stable bound solitons in a figure-eight fiber laser. , 2002, Optics letters.

[6]  Hie-Tae Moon,et al.  Soliton-kink interactions in a generalized nonlinear Schrödinger system , 2000 .

[7]  Yi-Tian Gao,et al.  Multi-Soliton and Rogue-Wave Solutions of the Higher-Order Hirota System for an Erbium-Doped Nonlinear Fiber , 2014 .

[8]  Wen-Xiu Ma,et al.  Solving the (3 + 1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm , 2012, Appl. Math. Comput..

[9]  Anjan Kundu,et al.  Landau-Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger-type equations , 1984 .

[10]  Wen-Xiu Ma,et al.  A coupled AKNS–Kaup–Newell soliton hierarchy , 1999 .

[11]  W. Eckhaus,et al.  Nonlinear evolution equations, rescalings, model PDEs and their integrability: II , 1987 .

[12]  David Artigas,et al.  Asymmetrical splitting of higher-order optical solitons induced by quintic nonlinearity , 1997 .

[13]  C. Dai,et al.  Bright and dark optical solitons in the nonlinear Schrodinger equation with fourth-order dispersion and cubic-quintic nonlinearity , 2005 .

[14]  Yuji Kodama,et al.  Solitons in optical communications , 1995 .

[15]  Yi-Tian Gao,et al.  Amplification of nonautonomous solitons in the Bose-Einstein condensates and nonlinear optics , 2011 .

[16]  Wen-Xiu Ma,et al.  A bilinear Bäcklund transformation of a (3+1) -dimensional generalized KP equation , 2012, Appl. Math. Lett..

[17]  Xiangzheng Li,et al.  The subsidiary ordinary differential equations and the exact solutions of the higher order dispersive nonlinear Schrödinger equation , 2006 .

[18]  R Grimm,et al.  Observation of Feshbach-like resonances in collisions between ultracold molecules. , 2005, Physical review letters.

[19]  A. Hasegawa.  Solitons in optical communications , 1995 .

[20]  P. G. Estévez.  Darboux transformation and solutions for an equation in 2+1 dimensions , 1999 .

[21]  Ying Liu,et al.  Soliton management for a variable-coefficient modified Korteweg-de Vries equation. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  Adrian Ankiewicz,et al.  Solitons : nonlinear pulses and beams , 1997 .

[23]  R. Hirota.  Exact solution of the Korteweg-deVries equation for multiple collision of solitons , 1971 .

[24]  Min Chen,et al.  Direct search for exact solutions to the nonlinear Schrödinger equation , 2009, Appl. Math. Comput..

[25]  P. G. Estevez,et al.  Darboux Transformations for a Bogoyavlenskii Equation in 2 + 1 Dimensions , 2000 .

[26]  Govind P. Agrawal,et al.  Nonlinear Fiber Optics , 1989 .

[27]  Wu Hong-Yu,et al.  Self-Similar Solutions of Variable-Coefficient Cubic-Quintic Nonlinear Schrödinger Equation with an External Potential , 2010 .

[28]  Bo Tian,et al.  Variable-coefficient higher-order nonlinear Schrödinger model in optical fibers: New transformation with burstons, brightons and symbolic computation , 2006 .

[29]  Peter A. Clarkson,et al.  Exact solutions of the multidimensional derivative nonlinear Schrodinger equation for many-body systems of criticality , 1990 .

[30]  Yi-Tian Gao,et al.  Dynamics of bound vector solitons induced by stochastic perturbations: Soliton breakup and soliton switching , 2013 .

[31]  Decio Levi,et al.  The Kundu–Eckhaus equation and its discretizations , 2009, 0904.4844.

[32]  Malomed.  Bound solitons in coupled nonlinear Schrödinger equations. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[33]  Junkichi Satsuma,et al.  Bilinearization of a Generalized Derivative Nonlinear Schrödinger Equation , 1995 .

[34]  M Lakshmanan,et al.  Coupled nonlinear Schrödinger equations with cubic-quintic nonlinearity: Integrability and soliton interaction in non-Kerr media. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[35]  Yi Qin,et al.  Bell-polynomial approach applied to the seventh-order Sawada-Kotera-Ito equation , 2014, Appl. Math. Comput..

[36]  Ewan M. Wright,et al.  Fundamental limit for integrated atom optics with Bose-Einstein condensates , 2003 .

[37]  Houria Triki,et al.  Exact analytic solitary wave solutions for the RKL model , 2009, Math. Comput. Simul..

[38]  A. H. Khater,et al.  Bäcklund Transformations and Exact Solutions for Alfvén Solitons in a Relativistic Electron–Positron Plasma , 1998 .

[39]  Wenxiu Ma,et al.  Bilinear Equations and Resonant Solutions Characterized by Bell Polynomials , 2013 .

[40]  C. De Angelis,et al.  Self-trapped propagation in the nonlinear cubic-quintic Schrodinger equation: a variational approach , 1994 .

[41]  Karlsson,et al.  Interactions between polarized soliton pulses in optical fibers: Exact solutions. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[42]  Enns,et al.  Bistable solitons. , 1986, Physical review letters.

[43]  Yi-Tian Gao,et al.  Multi-soliton solutions for the three-coupled KdV equations engendered by the Neumann system , 2014 .

[44]  Chen,et al.  Basic dynamic properties of the high-order nonlinear Schrödinger equation. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[45]  P L Chu,et al.  Breathing spatial solitons in non-Kerr media. , 1997, Optics letters.

[46]  Yi-Tian Gao,et al.  Nonautonomous matter waves in a spin-1 Bose-Einstein condensate. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[47]  F. B. Pelap,et al.  Solitonlike excitations in a one-dimensional electrical transmission line , 2005 .

[48]  J. Cardy,et al.  QUANTUM INVERSE SCATTERING METHOD AND CORRELATION FUNCTIONS , 1995 .

[49]  戴朝卿,et al.  Bright and dark optical solitons in the nonlinear SchrSdinger equation with fourth-order dispersion and cubic-quintic nonlinearity , 2005 .

[50]  Yuji Kodama,et al.  Optical solitons in a monomode fiber , 1985 .

[51]  T. Frederico,et al.  Stability of trapped Bose-Einstein condensates , 2001 .

[52]  Woo-Pyo Hong,et al.  Optical solitary wave solutions for the higher order nonlinear Schrödinger equation with cubic-quintic non-Kerr terms , 2001 .