Multiple exp-function method for soliton solutions of nonlinear evolution equations

We applied the multiple exp-function scheme to the(2+1)-dimensional Sawada-Kotera(SK) equation and(3+1)-dimensional nonlinear evolution equation and analytic particular solutions have been deduced. The analytic particular solutions contain one-soliton, two-soliton, and three-soliton type solutions. With the assistance of Maple, we demonstrated the efficiency and advantages of the procedure that generalizes Hirota's perturbation scheme. The obtained solutions can be used as a benchmark for numerical solutions and describe the physical phenomena behind the model.     

A new method for constructing soliton solutions and periodic solutions of nonlinear evolution equations

With the aid of the symbolic computation, we improve Xie's algorithm [F. Xie, Z.Y. Yan, H. Zhang, Phys. Lett. A 285 (2001) 76], and present a new extended method. Based on the new general ansatz (3), we successfully solve a compound KdV-MKdV equation, and obtain some special solutions which contain soliton solutions, and periodic solutions. The method can also be applied to other nonlinear partial differential equations.     

用三角函数法获得非线性Boussinesq方程的广义孤子解

找到一个合适的代换——三角函数法，将非线性Boussinesq微分方程转换为非线性代数方程组.用吴消元法求解该非线性代数方程组，从而获得一般形式Boussinesq微分方程的广义孤子解. 关键词： Boussinesq方程 吴消元法 非线性代数方程组 孤子解     

Double-Alekseev inverse scattering method in the stationary axi-symmetric vacuum gravitation field equations

We present a new improvement to the Alekseev inverse scattering method. This improved inverse scattering method is extended to a double form, followed by the generation of some new solutions of the double-complex Kinnersley equations. As the double-complex function method contains the Kramer-Neugebauer substitution and analytic continuation, a pair of real gravitation soliton solutions of the Einstein’s field equations can be obtained from a double N-soliton solution. In the case of the flat Minkowski space background solution, the general formulas of the new solutions are presented.      

Exact class of inverse scattering solutions

After presenting a procedure for finding exact solutions to the Gelfand-Levitan equation for inverse scattering, we present explicit solutions for 3- to 6-pole reflection coefficients. The method is applicable to an arbitrarily large number of poles. Here we give the first explicit treatment of 4, 5, and 6 poles.     

Yang-Mills equations as inverse scattering problem

Non-linear self-duality equations are shown to be conditions of compatibility of two linear equations. All the N-instanton fields are constructed explicitly.     

构造孤子方程的Weierstrass椭圆函数解的一个新方法

利用具有Weierstrass椭圆函数解的方程，首先获得了投影Riccati方程的两组新解.由于投影Riccati方程可用于多种具孤子解的非线性演化方程的求解，因而得到了一个可以构造这些方程的Weierstrass椭圆函数解的新方法. 关键词： Weierstrass椭圆函数解 投影Riccati方程 非线性演化方程     

Solutions of the anti-self-dualSU(2) Yang-Mills equations gained from the inverse scattering method

By applying the inverse scattering method in connection with a dressing procedure a huge class of solutions of the anti-self-dualSU(2) Yang-Mills equation is obtained. This class contains also the well-known't Hooft instantions.     

Pure soliton solutions of some nonlinear partial differential equations

A general approach is given to obtain the system of ordinary differential equations which determines the pure soliton solutions for the class of generalized Korteweg-de Vries equations (cf. [6]). This approach also leads to a system of ordinary differential equations for the pure soliton solutions of the sine-Gordon equation.     

On the method of inverse scattering problem and Bäcklund transformations for supersymmetric equations

Supersymmetric Liouville and sine-Gordon equations are studied. We write down for these models the system of linear equations for which the method of inverse scattering should be applicable. Expressions for an infinite set of conserved currents are explicitly given. Supersymmetric Bäcklund transformations and generalized conservation laws are constructed.     

A Reduced System of Linear Algebraic Equations for Giving Soliton Solutions of the Generalized ZS/AKNS Systems

A simple method for finding soliton solutions of the generaked ZS/AKNS systems whose Lax pairs are matrices with high orders is considered. An explicit expreesion of transformation between the Jost solution relating to the (n-1)-soliton solution and that relating to the n-soliton solution is found. A reduced system of N algebraic equations for giving N soliton solutions is deduced, it has an identical form no matter how high the order of matrices of the Lax pain is.     

Axially symmetric static solutions of four-dimensional SU(N) σ-models by the inverse scattering method

A systematic framework is derived for constructing a superpotential in static, axially symmetric four-dimensional SU(N) principal σ-models by applying an inverse scattering method.     

Supersymmetric Yang-Mills equations as an inverse scattering problem

The equations of motion (for N=3, 4) and the constraint equations (N=1, 2) for supersymmetric Yang-Mills theories are shown to be the compatibility conditions of some system of linear equations with a parameter.     

On the physical interpretation of some simple soliton solutions of Einstein's equations

The simple soliton solutions of Einstein's equations obtained by Belinski and Zakharov using the inverse scattering method have been interpreted as gravitational (solitary) shock waves, partly on the basis of an analysis of certain (coordinate) singularities apparently inherent to the method. A closer study reveals, however, that such singularities can be removed by an appropriate extension of the solutions, which is given explicitly. A classification of inequivalent flat space-time metrics appropriate for the applications of the method is derived. The problem of matching the Belinski-Zakharov (B-Z) simple solitons to flat space-time is analyzed and found to have more than one solution depending on the type of singularity admitted in the Ricci tensor. This is further illustrated by considering a three-parameter solution, inequivalent to that of Belinski and Zakharov. For negative values of one of these parameters the ranges of the coordinates are limited only by curvature singularities. For positive values of the parameter, coordinate singularities, similar to those in the B-Z solution, are also present. In this case, however, a matching to flat space-time leads to a shock front whose intersection with any spacelike hypersurface is bounded, in contrast with the behavior displayed by the B-Z solutions. The limiting case when the parameter is zero is found to have some special properties. A smooth extension is also shown to exist.This research was supported through a fellowship from the Consejo Nacional de Investigaciones Cientificas y Technicas de la Republica Argentina.     

两个非线性发展方程的双向孤波解与孤子解

采用分步确定拟解的原则, 对齐次平衡法求非线性发展方程孤子解的关键步骤作了进一步改 进. 以广义Boussinesq方程和bidirectional Kaup-Kupershmidt方程为应用实例, 说明使用 该方法可有效避免“中间表达式膨胀”的问题, 除获得标准Hirota形式的孤子解外, 还能获 得其他形式的孤子解. 关键词： 齐次平衡法 孤子解 孤波解 广义Boussinesq方程 bidirectional Kaup-Kupershmi dt方程     

Singular and non-topological soliton solutions for nonlinear fractional differential equations

In this article, the fractional derivatives are described in the modified Riemann–Liouville sense. We propose a new approach, namely an ansatz method, for solving fractional differential equations(FDEs) based on a fractional complex transform and apply it to solve nonlinear space–time fractional equations. As a result, the non-topological as well as the singular soliton solutions are obtained. This method can be suitable and more powerful for solving other kinds of nonlinear fractional FDEs arising in mathematical physics.     

Quantum inverse scattering method and the Heisenberg ferromagnet

The quantum version of the inverse scattering method is formulated for the Heisenberg ferromagnet. The elements of the transition matrix for the corresponding auxiliary linear problem are the quantum analogue of the action-angle variables. In a continuous limit the model is shown to yield the quantum nonlinear Schrödinger equation.     

非线性波方程尖峰孤子解的一种简便求法及其应用

根据尖峰孤子解的特点,提出了一种待定系数法求非线性波方程尖峰孤子解的思路和方法,并利用该方法求解了5个非线性波方程,即CH（Camassa-Holm)方程、五阶KdV-like 方程、广义Ostrovsky方程、组合KdV-mKdV方程和Klein-Gordon方程,比较简便地得到了这些方程的尖峰孤子解.文献中关于CH方程的结果成为本文结果的特例.通过数值模拟给出了部分解的图像.简要说明了非线性波方程存在尖峰孤子解所须满足的特定条件.该方法也适用于求其他非线性波方程的尖峰孤子解. 关键词： 非线性波方程 尖峰孤子解 待定系数法     

Numerical evidence for breakdown of soliton behaviour in solutions of the Maxwell-Bloch equations

Some results of a numerical study of colliding solitary wave solutions of the Maxwell-Bloch equations are reported. At high densities these pulses do not behave as solitons: on collision they undergo deformations and throw off oscillating tails.