Darboux Transformation and Variable Separation Approach： the Nizhnik-Novikov-Veselov Equation

Darboux transformation (DT) is developed to systematically find variable separation solutions for the Nizhnik-Novikov-Veselov equation. Starting from a seed solution with some arbitrary functions, the one-step DT yields the variable separable solutions, which can be obtained from the truncated Painleve analysis, and the two-step DT leads to some new variable separable solutions, which are the generalization of the known results obtained by using a guess ansatz to solve the generalized trilinear equation.     

Exact Solutions of a Fractional-Type Differential-Difference Equation Related to Discrete MKdV Equation

The extended simplest equation method is used to solve exactly a new differential-difference equation of fractional-type, proposed by Narita [J. Math. Anal. Appl. 381 （2011） 963] quite recently, related to the discrete MKdV equation. It is shown that the model supports three types of exact solutions with arbitrary parameters： hyperbolic, trigonometric and rational, which have not been reported before.     

Soliton Solutions of Bose-Einstein Condensate in Linear Magnetic Field and Time-Dependent Laser Field

With the help of symbolic computation, the tanh method is extended to find some new exact solutions of nonlinear Gross-Pitaevskii equation with weak bias magnetic and time-dependent laser fields. As a result, the bright and dark soliton solutions are obtained. In addition, some new soliton solutions in this model are found.     

Exact solutions of non-Hermitian chains with asymmetric long-range hopping under specific boundary conditions

We study the one-dimensional general non-Hermitian models with asymmetric long-range hopping and explore how to analytically solve the systems under some specific boundary conditions.Although the introduction of long-range hopping terms prevents us from finding analytical solutions for arbitrary boundary parameters,we identify the existence of exact solutions when the boundary parameters fulfill some constraint relations,which give the specific boundary conditions.Our analytical results show that the wave functions take simple forms and are independent of hopping range,while the eigenvalue spectra display rich model-dependent structures.Particularly,we find the existence of a special point coined as pseudo-periodic boundary condition,for which the eigenvalues are the same as those of the periodical system when the hopping parameters fulfill certain conditions,whereas the eigenstates display the non-Hermitian skin effect.     

Coherent soliton structures of the (2+1)-dimensional long-wave－short-wave resonance interaction equation

The variable separation approach is used to find exact solutions of the (2+1)-dimensional long-wave－short-wave resonance interaction equation. The abundance of the coherent soliton structures of this model is introduced by the entrance of an arbitrary function of the seed solutions. For some special selections of the arbitrary function, it is shown that the coherent soliton structures may be dromions, solitoffs, etc.     

Exact solutions and linear stability analysis for two-dimensional Ablowitz Ladik equation

The Ablowitz-Ladik equation is a very important model in nonlinear mathematical physics. In this paper, the hyper- bolic function solitary wave solutions, the trigonometric function periodic wave solutions, and the rational wave solutions with more arbitrary parameters of two-dimensional Ablowitz-Ladik equation are derived by using the （GI/G）-expansion method, and the effects of the parameters （including the coupling constant and other parameters） on the linear stability of the exact solutions are analysed and numerically simulated.     

Solitons for a generalized variable-coefficient nonlinear Schrdinger equation

In this paper,we investigate some exact soliton solutions for a generalized variable-coefficients nonlinear Schrdinger equation (NLS) with an arbitrary time-dependent linear potential which describes the dynamics of soliton solutions in quasi-one-dimensional Bose-Einstein condensations. Under some reasonable assumptions,one-soliton and two-soliton solutions are constructed analytically by the Hirota method. From our results,some previous one-and two-soliton solutions for some NLS-type equations can be recovered by some appropriate selection of the various parameters. Some figures are given to demonstrate some properties of the one-and the two-soliton and the discussion about the integrability property and the Hirota method is given finally.     

Solitons for a generalized variable-coefficient nonlinear SchrSdinger equation

In this paper, we investigate some exact soliton solutions for a generalized variable-coefficients nonlinear SchrSdinger equation （NLS） with an arbitrary time-dependent linear potential which describes the dynamics of soliton solutions in quasi-one-dimensional Bose-Einstein condensations. Under some reasonable assumptions, one-soliton and two-soliton solutions are constructed analytically by the Hirota method. From our results, some previous one- and two- soliton solutions for some NLS-type equations can be recovered by some appropriate selection of the various parameters. Some figures are given to demonstrate some properties of the one- and the two-soliton and the discussion about the integrability property and the Hirota method is given finally.     

A Series of Exact Solutions of （2＋l）-Dimensional CDGKS Equation

An algebraic method with symbolic computation is devised to uniformly construct a series of exact solutions of the （2＋1）-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawda equation. The solutions obtained in this paper include solitary wave solutions, rational solutions, triangular periodic solutions, Jacobi and Weierstrass doubly periodic solutions. Among them, the Jacobi periodic solutions exactly degenerate to the solutions at a certain limit condition. Compared with most existing tanh method, the method used here can give new and more general solutions. More importantly, this method provides a guldeline to classifj, the various types of the solution according to some parameters.     

Doubly Periodic Wave Solutions of Jaulent-Miodek Equations Using Variational Iteration Method Combined with Jacobian-function Method

One of the advantages of the variational iteration method is the free choice of initial guess. In this paper we use the basic idea of the Jacobian-function method to construct a generalized trial function with some unknown parameters. The Jaulent-Miodek equations are used to illustrate effectiveness and convenience of this method, some new explicit exact travelling wave solutions have been obtained, which include bell-type soliton solution, kink-type soliton solutions, solitary wave solutions, and doubly periodic wave solutions.     

New Exact Solutions of （1 ＋ 1）-Dimensional Coupled Integrable Dispersionless System

This paper applies the exp-function method, which was originally proposed to find new exact travelling wave solutions of nonlinear evolution equations, to the Riccati equation, and some exact solutions of this equation are obtained. Based on the Riccati equation and its exact solutions, we find new and more generalvariable separation solutions with two arbitrary functions of (1+1)-dimensional coupled integrable dispersionless system. As some special examples, some new solutions can degenerate into variable separation solutions reported in open literatures. By choosing suitably two independent variables p(x) and q(t) inour solutions, the annihilation phenomena of the flat-basin soliton, arch-basin soliton, and flat-top soliton are discussed.     

Persistence of travelling waves in a generalized Fisher equation

Travelling waves of the Fisher equation with arbitrary power of nonlinearity are studied in the presence of long-range diffusion. Using analogy between travelling waves and heteroclinic solutions of corresponding ODEs, we employ the geometric singular perturbation theory to prove the persistence of these waves when the influence of long-range effects is small. When the long-range diffusion coefficient becomes larger, the behaviour of travelling waves can only be studied numerically. In this case we find that starting with some values, solutions of the model lose monotonicity and become oscillatory.     

New Exact Solutions and Special Coherent Structures for Coupled Integrable Dispersionless Equation

Using improved homogeneous balance method, we obtain new exact solutions for the coupled integrable dispersionless equation. On the basis of these exact solutions, we find some new interesting coherent structures by selecting arbitrary functions appropriately.     

Localized Excitations of (2+1)-Dimensional Korteweg-de Vries System Derived from a Periodic Wave Solution

With the aid of an improved projective approach and a linear variable separation method,new types of variable separation solutions (including solitary wave solutions,periodic wave solutions,and rational function solutions)with arbitrary functions for (2 1)-dimensional Korteweg-de Vries system are derived.Usually,in terms of solitary wave solutions and rational function solutions,one can find some important localized excitations.However,based on the derived periodic wave solution in this paper,we find that some novel and significant localized coherent excitations such as dromions,peakons,stochastic fractal patterns,regular fractal patterns,chaotic line soliton patterns as well as chaotic patterns exist in the KdV system as considering appropriate boundary conditions and/or initial qualifications.     

α螺旋蛋白质螺旋链模型的精确解组

文章运用Maple语言程序,在没有假设的条件下,得到了α螺旋蛋白质螺旋链运动模型方程组的行波精确解组,它涵盖了所有的耦合解组与非耦合解组,具有任意性.耦合解组的算例函数及其特性分析,解释了α螺旋蛋白质螺旋链运动模型的行波孤立子解的耦合效应,揭示了增加、稳定和控制蛋白质活性和功能的方向,文章的研究方法,为求解生物大分子螺旋链运动模型的行波精确解组探索了溪径.     

Interband transitions of superlattices in arbitrary time-dependent electric fields: an analytical approach of closed-form solutions in the real space

Within a two-band tight-binding model driven by arbitrary time-dependent electric fields, we investigate the interband transitions and obtain closed-form solutions in the real space, from which we show that there are two essentially opposite evolution behaviors of the system: interband resonances and miniband localization. We also find that in weak interband coupling, as expected, the perturbative solutions are in good agreement with the exact numerical calculations.     

New Exact Solutions for （1 ＋ 1）-Dimensional Dispersion-Less System

Using improved homogeneous balance method, we obtain complex function form new exact solutions for the （1＋1）-dimensional dispersion-less system, and from the exact solutions we derive real function form solution of the field u. Based on this real function form solution, we find some new interesting coherent structures by selecting arbitrary functions appropriately.     

(2+1)维非线性薛定谔方程的环孤子,dromion,呼吸子和瞬子

利用分离变量法,研究了(2+1)维非线性薛定谔(NLS)方程的局域结构.由于在B?cklund变换和变量分离步骤中引入了作为种子解的任意函数,得到了NLS方程丰富的局域结构.合适地选择任意函数,局域解可以是dromion,环孤子,呼吸子和瞬子.dromion解不仅可以存在于直线孤子的交叉点上,也可以存在于曲线孤子的最近邻点上.呼吸子在幅度和形状上都进行了呼吸 关键词： 非线性薛定谔方程 分离变量法 孤子结构