MKdV方程的反散射解

本文考虑修正KdV(MKdV)方程u_t+6u~2u_x+u_(xxx)=0的反散射解,给出当反射系数为零且特征根为纯虚数时解的简洁表达式,并讨论了单孤子解和双孤子解。     

利用Darboux变换对KdV方程孤子解普遍性质的讨论

该文指出：利用Darboux变换不但可以非常简洁地得到文献［1］关于KdV方程单孤子解和双孤子解，而且便于讨论KdV方程的任意孤子解的性质．通过对KdV方程三孤子解的重点讨论，以及对KdV方程多孤子解的解析分析，得到了关于KdV方程任意阶孤子解的一些非常有意义的普遍结果．这些结果对于人们深入了解孤子相互作用规律具有重要的现实意义．     

高维扰动破裂孤子方程行波解的渐近解法

本文研究一类高维扰动破裂孤子方程.首先讨论对应典型的破裂方程,利用投射方法得到孤子精确行波解.再利用修改的广义投射近似方法得到扰动破裂孤子方程的行波渐近解.     

高维弱扰动破裂孤子波方程行波解

研究了一类高维弱扰动破裂孤子波方程.首先讨论了对应的典型破裂孤子波方程, 利用待定系数投射方法得到了孤子波精确解.再利用泛函分析和摄动理论得到了原弱扰动破裂孤子波方程的孤子行波渐近解.最后, 举出例子说明了用该方法得到的弱扰动破裂孤子波方程的行波渐近解具有简捷、有效和较高精度的优点.     

推广的（2＋1）维浅水波方程的精确解及其混沌行为

利用Riccati方程映射法和变量分离法,得到了推广的（2＋1）维浅水波系统的变量分离解（包括孤波解、周期波解和有理函数解）.根据得到的孤波解,构造出了方程的单孤子和双孤子结构,研究了孤子的混沌行为.     

（2＋1）维KP方程的三类精确解

研究了（2＋1）维KP方程的孤子解问题．应用Riccati方程映射法,得到了（2＋1）维KP方程的新的显式精确解的结构．根据得到的精确解结构,构造出了该方程的三类精确解．     

高维非线性K－G方程解之间的关系与N孤子解

设与Ｆｉ对应的解为共轭的正则动量为πｉ．场的正则变换ｇ：（πｉ）→（ｑ，ｐ）给出方程的一类解［ｑ（ｘ）］及诸间的关系，其中某些特解即为Ｇｉｂｂｏｎｊ和楼森岳等人几篇文章的主要结果．特别讨论了ｓＧ方程的一类解之间的递推关系，由此得到Ｎ个不同的孤子解，并分别给出３维和４维情形相应的Ｎ孤子特解．     

关于(2+1)-维色散的长波方程求解的一个注记

在文献[3]的基础上，根据一些简单方程的特征，导出了(2十1)—维色散的长波方程的新的精确解，其中包含了已有文献中的孤子解，多孤子解等．     

耦合KdV方程的几个精确解

Darboux变换是求孤子方程的精确解的一种新方法。它借助于孤子方程的Lax对。从方程的平凡解导出新的非平凡解。本文对一个四阶特征值问题找出了Darboux变换,并由此得到耦合KdV方程的孤子解,周期解,极点解等。     

关于KdV方程孤子解的研究

KdV方程的多孤子解很难直接验证，本文通过证明GLM反散射变换方程导出的Jost解满足两个Lax方程的方法，解决了这个问题．     

Sine-Gordon方程孤立子解的表达式

本文研究Sine-Gordon方程
u_xt=sinu(A)
的反散射解.给出了(A)的孤立子解的简洁表达式,并讨论了单孤立子解和双孤立子解.     

Reverse Space‐Time Nonlocal Sine‐Gordon/Sinh‐Gordon Equations with Nonzero Boundary Conditions

Nonlocal reverse space‐time Sine/Sinh‐Gordon type equations were recently introduced. They arise from a remarkably simple nonlocal reduction of the well‐known AKNS scattering problem, hence, they constitute an integrable evolution equations. Furthermore, the inverse scattering transform (IST) for rapidly decaying data was also constructed. In this paper, the IST for these novel nonlocal equations corresponding to nonzero boundary conditions (NZBCs) at infinity is presented. The NZBC problem is more complex due to the intricate branching structure of the associated linear eigenfunctions. Two cases are analyzed, which correspond to two different values of the phase at infinity. Special soliton solutions are discussed and explicit 1‐soliton and 2‐soliton solutions are found. Both spatially independent and spatially dependent boundary conditions are considered.     

(3+1)维时间分数阶KdV-Zakharov-Kuznetsov方程的分支分析及其行波解

首先,运用拟设方法和动力系统分支方法,获得了(3+1)维时间分数阶KdV-Zakharov-Kuznetsov方程的奇异孤子解、 亮孤子解、 拓扑孤子解、 周期爆破波解、 孤立波解等.再利用MAPLE软件画出了KdV-Zakharov-Kuznetsov方程在不同条件下的分支相图.最后,讨论了行波解之间的联系.     

New travelling wave solutions for a nonlinearly dispersive wave equation of Camassa-Holm equation type

In this paper, the integral bifurcation method is used to study a nonlinearly dispersive wave equation of Camassa-Holm equation type. Loop soliton solution and periodic loop soliton solution, solitary wave solution and solitary cusp wave solution, smooth periodic wave solution and non-smooth periodic wave solution of this equation are obtained, their dynamic characters are discussed. Some solutions have an interesting phenomenon that one solution admits multi-waves when parameters vary.     

Non-linear partial differential equations via vector fields on homogeneous Banach manifolds

The vector field formulation of and the Sato-Segal-Wilson approach to soliton equations are related to each other in this paper. From Banach Lie groups associated with the MKdV hierarchy of differential equations, we derive homogeneous Banach manifolds of solutions on which these equations are realized by vector fields. In the same way, we obtain homogeneous Banach manifolds of solutions to the sine-Gordon equation. The scattering and inverse scattering maps in this set-up are also discussed.     

Dark solitons in N-coupled higher order nonlinear Schrödinger equations

By considering the third order dispersion, self-steepening and stimulated Raman scattering effects, we analyse the dark soliton propagation in N-coupled higher order nonlinear Schrödinger equations. Using Painlevé analysis, we prove that this system is completely integrable. The result is confirmed further by the presentation of Lax pair. Using the Hirota method, the construction of soliton solution is discussed.     

Solitons, chaos and fractals in the (2 + 1)-dimensional dispersive long wave equation

For a higher-dimensional integrable nonlinear dynamical system, there are abundant coherent soliton excitations. With the aid of a projective Riccati equation approach, the paper obtains several types of exact solutions to the (2 + 1)-dimensional dispersive long wave (DLW) equation which include multiple soliton solution, periodic soliton solution and Weierstrass function solution. Subsequently, several multisolitons are derived and some novel features are revealed by introducing lower-dimensional patterns.     

Explicit Complex-Valued Solutions of the Korteweg–de Vries Equation on the Half-Line and on the Whole-Line

The paper deals with the initial-value problems for the Korteweg–de Vries (KdV) equations on the half-line and on the whole-line for complex-valued measurable and exponentially decreasing potentials. The time evolution equation for the reflection coefficient is derived and then a one-to-one correspondence between the scattering data and the solution of the KdV equation is shown. Families of exact solutions of the KdV equation are represented for the class of reflection-free potentials, in which the inverse scattering problem associated with the KdV equation can be solved exactly. Some helpful examples of soliton solutions of the KdV equation are provided.     

Dynamics of a coupled modified (1 + 1)‐dimensional Toda equation of BKP type

In this paper, we present a new coupled modified (1 + 1)‐dimensional Toda equation of BKP type (Kadomtsev‐Petviashvilli equation of B‐type), which is a reduction of the (2 + 1)‐dimensional Toda equation. Two‐soliton and three‐soliton solutions to the coupled system are derived. Furthermore, the N‐soliton solution is presented in the form of Pfaffian. The asymptotic analysis of two‐soliton solutions is studied to explain their collision properties. It is shown that the coupled system exhibit richer interaction phenomena including soliton fission, fusion, and mixed collision. Copyright © 2015 John Wiley & Sons, Ltd.