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．     

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．     

四耦合非线性薛定谔方程的矢量孤子解及叠加解

本文通过两种方法分别得到四耦合非线性薛定谔方程的矢量孤子解及一阶叠加解.第一种方法是利用发展的广田双线性方法,得到四耦合非线性薛定谔方程的单、双孤子解,以及一种具有呼吸行为的新解.第二种方法是利用一阶达布变换,得到一阶怪波解以及怪波与孤子、呼吸子相互作用的一阶叠加解.     

Broer-Kaup系统的达布变换及其孤子解

根据Broer-Kaup系统的Lax对, 借助Broer-Kaup系统的谱问题的规范变换, 一个包含多参数的达布变换被构造出. 以一个平凡解作为种子解, 利用达布变换, 可以求得Broer-Kaup系统的非平凡解的一般表达式. 并且讨论了N=1和N=2两种孤子解的情形. 这是一种与2X2谱问题有关的孤子碰撞图像的新类型.     

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.     

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.     

用Hirota双线性方法构造一种(3+1)维高维孤子方程的多孤子解

通过两种方法构造了一种(3+1)维高维孤子方程的孤子解.第一种方法是利用对数函数变换,将其化成双线性形式的方程,在用级数扰动法求解双线性方程的单孤子解、双孤子解和N-孤子解.第二种方法是用广义有理多项式与试探法相结合,构造了(3+1)维高维孤子方程的怪波解.     

散焦mKdV方程的N重暗孤子解的行列式表示

该文首先构造了耦合的mKdV方程的新的达布变换,同时显式给出了它的达布矩阵T_N和新解q~([N]),r~([N])的行列式表示.其次,考虑将约化条件r=q*附加到该达布变换上,以及考虑一个周期的非零种子解,得到了散焦mKdV方程的N重暗孤子解的行列式表示.最后,证明了暗的单孤子解和暗的2孤子解是光滑的,进一步证明了暗的N(N≥2)孤子解至少在某一邻域内是光滑的.     

广义高维Klein-Gordon强迫扰动方程的孤子解

利用同伦映射方法研究了一类非线性广义强迫扰动Klein-Gordon方程.首先利用双曲正切待定系数法求得了无扰动项典型方程的孤子解.然后利用同伦映射原理得到了强迫扰动Klein-Gordon方程的任意次近似孤子解.最后叙述了得到的近似孤子解是一个解析展开式,还能对它进行解析运算.这对使用简单的模拟方法得到的近似解是达不到的.     

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.     

A generalized AKNS hierarchy,bi-Hamiltonian structure,and Darboux transformation

A new generalized AKNS hierarchy is presented starting from a 4 × 4 matrix spectral problem with four potentials. Its generalized bi-Hamiltonian structure is also investigated by using the trace identity. Moreover, the special coupled nonlinear equation, the coupled KdV equation, the KdV equation, the coupled mKdV equation and the mKdV equation are produced from the generalized AKNS hierarchy. Most importantly, a Darboux transformation for the generalized AKNS hierarchy is established with the aid of the gauge transformation between the corresponding 4 × 4 matrix spectral problem, by which multiple soliton solutions of the generalized AKNS hierarchy are obtained. As a reduction, a Darboux transformation of the mKdV equation and its new analytical positon, negaton and complexiton solutions are given.     

Darboux transformations for a matrix long-wave–short-wave equation and higher-order rational rogue-wave solutions

A new matrix long-wave–short-wave equation is proposed with the of help of the zero-curvature equation. Based on the gauge transformation between Lax pairs, both onefold and multifold classical Darboux transformations are constructed for the matrix long-wave–short-wave equation. Resorting to the classical Darboux transformation, a multifold generalized Darboux transformation of the matrix long-wave–short-wave equation is derived by utilizing the limit technique, from which rogue wave solutions, in particular, can be obtained by employing the generalized Darboux transformation. As applications, we obtain rogue-wave solutions of the long-wave–short-wave equation and some explicit solutions of the three-component long-wave–short-wave model, including soliton solutions, breather solutions, the first-order and higher-order rogue-wave solutions, and others by using the generalized Darboux transformation.     

Solving soliton equations with self-consistent sources by method of variation of parameters

Starting from the solutions of soliton equations and corresponding eigenfunctions obtained by Darboux transformation, we present a new method to solve soliton equations with self-consistent sources (SESCS) based on method of variation of parameters. The KdV equation with self-consistent sources (KdVSCS) is used as a model to illustrate this new method. In addition, we apply this method to construct some new solutions of the derivative nonlinear Schrödinger equation with self-consistent sources (DNLSSCS) such as phase solution, dark soliton solution, bright soliton solution and breather-type solution.     

Bell polynomial approach to an extended Korteweg–de Vries equation

Under investigation in this paper is an extended Korteweg–de Vries equation. Via Bell polynomial approach and symbolic computation, this equation is transformed into two kinds of bilinear equations by choosing different coefficients, namely KdV–SK‐type equation and KdV–Lax‐type equation. On the one hand, N‐soliton solutions, bilinear Bäcklund transformation, Lax pair, Darboux covariant Lax pair, and infinite conservation laws of the KdV–Lax‐type equation are constructed. On the other hand, on the basis of Hirota bilinear method and Riemann theta function, quasiperiodic wave solution of the KdV–SK‐type equation is also presented, and the exact relation between the one periodic wave solution and the one soliton solution is established. It is rigorously shown that the one periodic wave solution tend to the one soliton solution under a small amplitude limit. Copyright © 2013 John Wiley & Sons, Ltd.     

Darboux transformation and explicit solutions for the Satsuma-Hirota coupled equation

A Darboux transformation for the Satsuma-Hirota coupled equation is obtained with the help of the gauge transformation between the Lax pairs. As an application of the Darboux transformation, we give some new explicit solutions, including rational solutions, soliton solutions and periodic solutions and others, of the Satsuma-Hirota coupled equation.     

矩阵AKNS系列Darboux变换的行列式表示

本文将建立矩阵AKNS系列Darboux变换的行列式表示。为此,推广了Sylvester恒等式,并利用它化简Darboux迭带所致的行列式 最后,给出了几个著名的矩阵孤立子方程,如矩阵KdV、矩阵NLS、矩阵MKdV等的孤立子解。     

Some new integrable systems constructed from the bi-Hamiltonian systems with pure differential Hamiltonian operators

When both Hamiltonian operators of a bi-Hamiltonian system are pure differential operators, we show that the generalized Kupershmidt deformation (GKD) developed from the Kupershmidt deformation in [10] offers an useful way to construct new integrable system starting from the bi-Hamiltonian system. We construct some new integrable systems by means of the generalized Kupershmidt deformation in the cases of Harry Dym hierarchy, classical Boussinesq hierarchy and coupled KdV hierarchy. We show that the GKD of Harry Dym equation, GKD of classical Boussinesq equation and GKD of coupled KdV equation are equivalent to the new integrable Rosochatius deformations of these soliton equations with self-consistent sources. We present the Lax pair for these new systems. Therefore the generalized Kupershmidt deformation provides a new way to construct new integrable systems from bi-Hamiltonian systems and also offers a new approach to obtain the Rosochatius deformation of soliton equation with self-consistent sources.     

Prolongation structures of a generalized coupled Korteweg-de Vries equation and Miura transformation

In this paper, the prolongation structures of a generalized coupled Korteweg-de Vries (KdV) equation are investigated and two integrable coupled KdV equations associated with their Lax pairs are derived. Furthermore, a Miura transformation related to a integrable coupled KdV equation is derived, from which a new coupled modified KdV equations is obtained.