Novel exact solutions of coupled nonlinear Schro¨dinger equations with time–space modulation

We construct various novel exact solutions of two coupled dynamical nonlinear Schrdinger equations.Based on the similarity transformation,we reduce the coupled nonlinear Schrdinger equations with time-and space-dependent potentials,nonlinearities,and gain or loss to the coupled dynamical nonlinear Schrdinger equations.Some special types of non-travelling wave solutions,such as periodic,resonant,and quasiperiodically oscillating solitons,are used to exhibit the wave propagations by choosing some arbitrary functions.Our results show that the number of the localized wave of one component is always twice that of the other one.In addition,the stability analysis of the solutions is discussed numerically.     

Solutions of Two Kinds of Non-Isospectral Generalized Nonlinear Schrodinger Equation Related to Bose-Einstein Condensates

Two non-isospectral generalized nonlinear Schrodinger （ONLS） equations, which are two important models of nonlinear excitations of matter waves in Bose-Einstein condensates, are studied. Two novel transformations are constructed such that these two GNLS equations are transformed to the well-known nonlinear Schr6dinger （NLS） equation, which is an isospectral equation. Therefore, once one solution of the NLS equation is provided, we can immediately obtain one solution for two ONLS equations by these transformations. Thus it is unnecessary to solve these two non-isospectral GNLS equations directly. Soliton solutions and periodic solutions are obtained for them by two transformations from the corresponding solutions of the NLS equation, which are generated by Darboux transformation.     

A high order energy preserving scheme for the strongly coupled nonlinear Schrōdinger system

A high order energy preserving scheme for a strongly coupled nonlinear Schrōdinger system is roposed by using the average vector field method. The high order energy preserving scheme is applied to simulate the soliton evolution of the strongly coupled Schrōdinger system. Numerical results show that the high order energy preserving scheme can well simulate the soliton evolution, moreover, it preserves the discrete energy of the strongly coupled nonlinear Schrōdinger system exactly.     

Higher-Order Localized Waves in Coupled Nonlinear Schrodinger Equations

Higher-order localized waves in coupled nonlinear Schr6dinger equations are investigated by the generalized Darboux transformation. We show that two dark-bright solitons together with a second-order rogue wave of fundamental or triangular pattern and two breathers together with a second-order rogue wave of fundamentM or triangular pattern coexist in the second-order localized wave for the coupled system. Moreover, by increasing the value of one free parameter, the nonlinear waves in the second-order localized wave can merge with each other. The results further reveal the abundant dynamic behaviors of localized waves in coupled systems.     

Soliton and rogue wave solutions of two-component nonlinear Schr?dinger equation coupled to the Boussinesq equation

The nonlinear Schr?dinger(NLS) equation and Boussinesq equation are two very important integrable equations.They have widely physical applications. In this paper, we investigate a nonlinear system, which is the two-component NLS equation coupled to the Boussinesq equation. We obtain the bright–bright, bright–dark, and dark–dark soliton solutions to the nonlinear system. We discuss the collision between two solitons. We observe that the collision of bright–bright soliton is inelastic and two solitons oscillating periodically can happen in the two parallel-traveling bright–bright or bright–dark soliton solution. The general breather and rogue wave solutions are also given. Our results show again that there are more abundant dynamical properties for multi-component nonlinear systems.     

Exact Soliton Solutions for the (2+1)-Dimensional Coupled Higher-Order Nonlinear Schr¨odinger Equations in Birefringent Optical-Fiber Communication

In birefringent optical fibers, the propagation of femtosecond soliton pulses is described by coupled higherorder nonlinear Schrdinger equations. In this paper, we will investigate the bright and dark soliton solutions of(2+1)-dimensional coupled higher-order nonlinear Schrdinger equations, with the aid of symbolic computation and the Hirota method. On the basis of soliton solutions, we test and discuss the interactions graphically between the solitons in the x-z, x-t, and z-t planes.     

New Doubly Periodic Solutions of Nonlinear Evolution Equations via Weierstrass Elliptic Function Expansion Algorithm

A Weierstrass elliptic function expansion method and its algorithm are developed in this paper. The method changes the problem solving nonlinear evolution equations into another one solving the corresponding system of nonlinear algebraic equations. With the aid of symbolic computation (e.g. Maple), the method is applied to the combined KdV-mKdV equation and (2 1)-dimensional coupled Davey-Stewartson equation. As a consequence, many new types of doubly periodic solutions are obtained in terms of the Weierstrass elliptic function. Jacobi elliptic function solutions and solitary wave solutions are also given as simple limits of doubly periodic solutions.     

Periodic solutions of nonlinear Schro^edinger type equations

By using the solutions of an auxiliary elliptic equation, a direct algebraic method is proposed to construct the exact solutions of nonlinear Schrfdinger type equations. It is shown that many exact periodic solutions of some nonlinear Schro^edinger type equations are explicitly obtained with the aid of symbolic computation, including corresponding envelope solitary and shock wave solutions.     

Trial function method and exact solutions to the generalized nonlinear Schrdinger equation with time-dependent coefficient

In this paper,the trial function method is extended to study the generalized nonlinear Schrdinger equation with timedependent coefficients.On the basis of a generalized traveling wave transformation and a trial function,we investigate the exact envelope traveling wave solutions of the generalized nonlinear Schrdinger equation with time-dependent coefficients.Taking advantage of solutions to trial function,we successfully obtain exact solutions for the generalized nonlinear Schrdinger equation with time-dependent coefficients under constraint conditions.     

Effect of Time-Dependent Atomic Scattering Length on Solitons in Bose-Einstein Condensates with a Complex Potential

We consider the one-dimensional nonlinear Schrǒdinger equations that describe the dynamics of a Bose-Einstein Condensates with time-dependent scattering length in a complex potential. Our results show that as long as the integrable relation is satisfied, exact solutions of the one-dimensional nonlinear Schrǒdinger equation can be found in a general closed form, and interactions between two solitons are modulated in a complex potential We find that the changes of the scattering length and trapping potential can be effectively used to control the interaction between two bright soliton.     

Variable Separation Solution for （1＋1）-Dimensional Nonlinear Models Related to Schroedinger Equation

A variable separation approach is proposed and successfully extended to the (1 1)-dimensional physics models. The new exact solution of (1 1)-dimensional nonlinear models related to Schr6dinger equation by the entrance of three arbitrary functions is obtained. Some special types of soliton wave solutions such as multi-soliton wave solution,non-stable soliton solution, oscillating soliton solution, and periodic soliton solutions are discussed by selecting the arbitrary functions appropriately.     

Multi-soliton solutions for the coupled modified nonlinear Schrdinger equations via Riemann–Hilbert approach

The coupled modified nonlinear Schrdinger equations are under investigation in this work. Starting from analyzing the spectral problem of the Lax pair, a Riemann–Hilbert problem for the coupled modified nonlinear Schrdinger equations is formulated. And then, through solving the obtained Riemann–Hilbert problem under the conditions of irregularity and reflectionless case, N-soliton solutions for the equations are presented. Furthermore, the localized structures and dynamic behaviors of the one-soliton solution are shown graphically.     

Rogue-wave pair and dark-bright-rogue wave solutions of the coupled Hirota equations

Novel explicit rogue wave solutions of the coupled Hirota equations are obtained by using the Darboux transformation.In contrast to the fundamental Peregrine solitons and dark rogue waves, we present an interesting rogue-wave pair that involves four zero-amplitude holes for the coupled Hirota equations. It is significant that the corresponding expressions of the rogue-wave pair solutions contain polynomials of the fourth order rather than the second order. Moreover, dark-brightrogue wave solutions of the coupled Hirota equations are given, and interactions between Peregrine solitons and dark-bright solitons are analyzed. The results further reveal the dynamical properties of rogue waves for the coupled Hirota equations.     

An Extended Method for Constructing Travelling Wave Solutions to Nonlinear Partial Differential Equations

In this paper, an extended method is proposed for constructing new forms of exact travelling wave solutions to nonlinear partial differential equations by making a more general transformation. For illustration, we apply the method to the asymmetric Nizhnik-Novikov-Vesselov equation and the coupled Drinfel＇d-Sokolov-Wilson equation and successfully cover the previously known travelling wave solutions found by Chen＇s method .     

Exact Soliton Solutions to a Generalized Nonlinear Schrodinger Equation

The （1＋1）-dimensional F-expansion technique and the homogeneous nonlinear balance principle have been generalized and applied for solving exact solutions to a general （3＋1）-dimensional nonlinear Schr6dinger equation （NLSE） with varying coefficients and a harmonica potential. We found that there exist two kinds of soliton solutions. The evolution features of exact solutions have been numerically studied. The （3＋1）D soliton solutions may help us to understand the nonlinear wave propagation in the nonlinear media such as classical optical waves and the matter waves of the Bose-Einstein condensates.     

（G′/G）-Expansion Method Equivalent to Extended Tanh Function Method

In a recent article [Physics Letters A 372 （2008） 417], Wang et al. proposed a method, which is called the （G′/G）-expansion method, to look for travelling wave solutions of nonlinear evolution equations. The travelling wave solutions involving parameters of the KdV equation, the mKdV equation, the variant Boussinesq equations, and the Hirota-Satsuma equations are obtained by using this method. They think the （G′/G）-expansion method is a new method and more travelling wave solutions of many nonlinear evolution equations can be obtained. In this paper, we will show that the （G′/G）-expansion method is equivalent to the extended tanh function method.     

Qualitative Analysis and Explicit Exact Solitary,Kink and Anti-Kink Wave Solutions of the Generalized Nonlinear Schrdinger Equation with Parabolic Law Nonlinearity

The generalized nonlinear Schrdinger equation with parabolic law nonlinearity is studied by using the factorization technique and the method of dynamical systems.From a dynamic point of view,the existence of smooth solitary wave,kink and anti-kink wave is proved and the sufficient conditions to guarantee the existence of the above solutions in different regions of the parametric space are given.Also,all possible explicit exact parametric representations of the waves are presented.     

Applications of the first integral method to nonlinear evolution equations

<正>In this paper,we establish travelling wave solutions for some nonlinear evolution equations.The first integral method is used to construct the travelling wave solutions of the modified Benjamin-Bona-Mahony and the coupled Klein-Gordon equations.The obtained results include periodic and solitary wave solutions.The first integral method presents a wider applicability to handling nonlinear wave equations.     

Exact periodic wave and soliton solutions in two-component Bose--Einstein condensates

We present several families of exact solutions to a system of coupled nonlinear Schr\"{o}dinger equations. The model describes a binary mixture of two Bose--Einstein condensates in a magnetic trap potential. Using a mapping deformation method, we find exact periodic wave and soliton solutions, including bright and dark soliton pairs.     

A New Method for Constructing Travelling Wave Solutions to the modified Benjamin--Bona--Mahoney Equation

We present a new method to find the exact travelling wave solutions of nonlinear evolution equations, with the aid of the symbolic computation. Based on this method, we successfully solve the modified BenjaminBona-Mahoney equation, and obtain some new solutions which can be expressed by trigonometric functions and hyperbolic functions, It is shown that the proposed method is direct, effective and can be used for many other nonlinear evolution equations in mathematical physics.