Generalized Darboux Transformation and Rational Solutions for the Nonlocal Nonlinear Schrodinger Equation with the Self-Induced Parity-Time Symmetric Potential

In this paper, I construct a generalized Darboux transformation for the nonlocal nonlinear Schrodinger equation with the self-induced parity-time symmetric potential. The N-order rational solution is derived by the iterative rule and it can be expressed by the determinant form. In particular, I calculate first-order and second-order rational solutions and obtain their figures according to different parameters.

Localized waves in three-component coupled nonlinear Schrdinger equation

We study the generalized Darboux transformation to the three-component coupled nonlinear Schr ¨odinger equation.First-and second-order localized waves are obtained by this technique.In first-order localized wave,we get the interactional solutions between first-order rogue wave and one-dark,one-bright soliton respectively.Meanwhile,the interactional solutions between one-breather and first-order rogue wave are also given.In second-order localized wave,one-dark-one-bright soliton together with second-order rogue wave is presented in the first component,and two-bright soliton together with second-order rogue wave are gained respectively in the other two components.Besides,we observe second-order rogue wave together with one-breather in three components.Moreover,by increasing the absolute values of two free parameters,the nonlinear waves merge with each other distinctly.These results further reveal the interesting dynamic structures of localized waves in the three-component coupled system.

Localized waves of the coupled cubic–quintic nonlinear Schrdinger equations in nonlinear optics

We investigate some novel localized waves on the plane wave background in the coupled cubic-quintic nonlinear Schrdinger （CCQNLS） equations through the generalized Darboux transformation （DT）. A special vector solution of the Lax pair of the CCQNLS system is elaborately constructed, based on the vector solution, various types of higher-order localized wave solutions of the CCQNLS system are constructed via the generalized DT. These abundant and novel localized waves constructed in the CCQNLS system include higher-order rogue waves, higher-order rogues interacting with multi-soliton or multi-breather separately. The first-and second-order semi-rational localized waves including several free parameters are mainly discussed：（i） the semi-rational solutions degenerate to the first-and second-order vector rogue wave solutions; （ii） hybrid solutions between a first-order rogue wave and a dark or bright soliton, a second-order rogue wave and two dark or bright solitons; （iii） hybrid solutions between a first-order rogue wave and a breather, a second-order rogue wave and two breathers. Some interesting and appealing dynamic properties of these types of localized waves are demonstrated, for example, these nonlinear waves merge with each other markedly by increasing the absolute value of α. These results further uncover some striking dynamic structures in the CCQNLS system.

含有混合导数项耦合非线性薛定谔方程高阶孤子解的动力学分析

本文利用广义达布变换研究了含有混合导数项耦合非线性薛定谔方程高阶孤子的动力学特性.基于Lax对方程的一对线性无关解,利用代数迭代过程推导出该方程组的单孤子解到三孤子解.通过数值模拟,研究该方程组的二孤子和三孤子,给出相应的动力学演化图,展现多孤子的动力学性态–孤子的弹性碰撞、非弹性碰撞和束缚态.研究表明,多孤子呈现出一些新颖的相互作用方式,这将有助于实验中对光孤子的进一步研究.

Integrability, multi-soliton and rational solutions, and dynamical analysis for a relativistic Toda lattice system with one perturbation parameter

Under investigation in this paper is a relativistic Toda lattice system with one perturbation parameterαabbreviated as RTLαsystem by Suris,which may describe the motions of particles in lattices interacting through an exponential interaction force.First of all,an integrable lattice hierarchy associated with an RTLαsystem is constructed,from which some relevant integrable properties such as Hamiltonian structures,Liouville integrability and conservation laws are investigated.Secondly,the discrete generalized(m,2 N-m)-fold Darboux transformation is constructed to derive multi-soliton solutions,higher-order rational and semirational solutions,and their mixed solutions of an RTLαsystem.The soliton elastic interactions and details of rational solutions are analyzed via the graphics and asymptotic analysis.Finally,soliton dynamical evolutions are investigated via numerical simulations,showing that a small noise has very little effect on the soliton propagation.These results may provide new insight into nonlinear lattice dynamics described by RTLαsystem.