广义耦合KdV孤子方程的达布变换及其奇孤子解

借助谱问题的规范变换, 给出广义耦合KdV孤子方程的达布变换,利用达布变换来产生广义耦合KdV孤子方程的奇孤子解,并且用行列式的形式来表达广义耦合KdV孤子方程的奇孤子解．作为应用,广义耦合KdV孤子方程奇孤子解的前两个例子被给出．     

广义耦合KdV孤子方程的达布变换及其偶孤子解

根据广义耦合KdV孤子方程的Lax对, 借助谱问题的规范变换, 一个包含多参数的达布变换被构造出来. 利用达布变换来产生广义耦合KdV孤子方程的偶孤子解, 并且用行列式的形式来表达广义耦合KdV孤子方程的偶孤子解. 作为应用, 广义耦合KdV孤子方程的偶孤子解的前两个例子被给出.     

一个求孤子方程有限带势解的方法

基于Lax对非线性化方法，我们以KdV方程为例给出了一个构造孤子方程的有限带势解的方法．通过Lax对非线性化KdV方程被分解成两个有限维可积系统，进而找到这些有限维可积系统公共的角-作用坐标，最终我们获得了KdV方程的有限带势解．     

超对称柱KdV方程的孤子解

利用直接法将柱KdV方程超对称化.通过适当的变换,利用双线性方法将超对称柱KdV方程双线性化,由超对称Hirota双线性导数法构造出超对称柱KdV方程的单孤子解、双孤子解、三孤子解以及n孤子解的具体表达形式.     

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．     

双模Jordan KdV方程的多孤子解与精确解

根据简化的Hirota双线性方法和cole-hopf变换,当双模Jordan KdV方程中的非线性参数与线性参数取特殊值时,得到了双模Jordan KdV方程的多孤子解.同时,当方程中非线性参数与线性参数取一般值,也得到了这个方程的其它的精确解.     

推广的F -展开法及变系数KdV和mKdV的精确解

该文首先推广了新近提出的F -展开法,利用该方法导出了变系数KdV和mKdV方程 的类椭圆函数解;并在极限的情况下,得到变系数KdV和 mKdV方程变波速和变波长的类孤子解以及其他形式解.     

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。     

随机广义KdV方程的随机精确解

通过一个辅助性方程和埃米尔特变换研究广义随机KdV方程的随机雅克比椭圆函数类波解.此外,还通过椭圆函数在模数取极限m→0和m→1的情况,给出方程的随机类孤子解和随机三角函数波解,所得结果丰富了广义随机KdV方程的精确解.     

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.     

A new integrable equation that combines the KdV equation with the negative‐order KdV equation

In this work, we develop a new integrable equation by combining the KdV equation and the negative‐order KdV equation. We use concurrently the KdV recursion operator and the inverse KdV recursion operator to construct this new integrable equation. We show that this equation nicely passes the Painlevé test. As a result, multiple soliton solutions and other soliton and periodic solutions are guaranteed and formally derived.     

A study on the (2 + 1)‐dimensional KdV4 equation derived by using the KdV recursion operator

We derive a new ( 2 + 1)‐dimensional Korteweg–de Vries 4 (KdV4) equation by using the recursion operator of the KdV equation. This study shows that the new KdV4 equation possess multiple soliton solutions the same as the multiple soliton solutions of the KdV hierarchy, but differ only in the dispersion relations. We also derive other traveling wave solutions. Copyright © 2013 John Wiley & Sons, Ltd.     

A note on “The integrable KdV6 equation: Multiple soliton solutions and multiple singular soliton solutions”

The goal of this short note is to provide another kind soliton solutions with Hirota form, which is different from what Wazwaz obtained in [A.M. Wazwaz, The integrable KdV6 equations: Multiple soliton solutions and multiple singular soliton solutions, Appl. Math. Comput. 204 (2008) 963-972]. Meanwhile we newly construct the MKdV6 equation and derive a Miura transformation between KdV6 equation and MKdV6 equation.     

A study on KdV and Gardner equations with time-dependent coefficients and forcing terms

In this work we study the KdV equation and the Gardner equation with time-dependent coefficients and forcing term for each equation. A generalized wave transformation is used to convert each equation to a homogeneous equation. The soliton ansatz will be applied to the homogeneous equations to obtain soliton solutions.     

New Solutions to the Ultradiscrete Soliton Equations

New solutions to the ultradiscrete soliton equations, such as the Box–Ball system, the Toda equation, etc. are obtained. One of the new solutions which we call a "negative-soliton" satisfies the ultradiscrete KdV equation (Box–Ball system) but there is not a corresponding traveling wave solution for the discrete KdV equation. The other one which we call a "static-soliton" satisfies the ultradiscrete Toda equation but there is not a corresponding traveling wave solution for the discrete Toda equation. A collision of a soliton with a negative-soliton generates many balls in a box over the capacity of the box in the Box–Ball system, while a collision of a soliton with the static-soliton describes, in the ultradiscrete limit, transmission of a soliton through junctions of a "nonuniform Toda equation." We have obtained exact solutions describing these phenomena.     

Soliton solutions for a generalized KdV and BBM equations with time-dependent coefficients

A generalized KdV equation with time-dependent coefficients will be studied. The BBM equation with time-dependent coefficients and linear damping term will also be examined. The wave soliton ansatz will be used to obtain soliton solutions for both equations. The conditions of existence of solitons are presented.     

EXACT SOLUTIONS OF THE VARIABLE COEFFICIENT KdV AND SG TYPE EQUATIONS

In this paper,the variable cofficient KdV equation with dissipative loss and nonuniformity terms and the variable coefficient SG equation with nonuniformity term are studied. The exact solutions of the KdV and SG equations are obtained. In particular,the soliton solutions oftwo equations are found.