广义色散Camassa-Holm模型的奇异孤子

引进一类广义色散Camassa-Holm模型，对其做奇异性分析.通过改进的WTC-Kruskal算法，证明该模型在Painlevé意义下可积，得到了它的一组Painlevé-Bcklund系统和Bcklund变换.应用Maple进行代数运算，得到了丰富的规则(regular)孤子和一类奇异(singular)孤子，扭结(kink)孤子,紧孤子(compacton)和反紧孤子(anti-compacton).特别地，推导出一类在扭结孤子的中间区域包含有一列周期尖点(cuspon)波的奇异结构.在这些规则的孤子系统的基础上，对可积广义系统应用Bcklund变换，得到三类奇异孤子，分别是具有驼峰结构的周期爆破波，具有爆破波结构的扭结孤子和紧孤子. 关键词： 广义Camassa-Holm 模型 周期尖点波 紧孤子 周期爆破波     

具有广义Virasoro对称代数的(3+1)维Painlevé可积模型

寻找高维可积模型(特别是3+1维可积模型)是非线性物理中的一个非常重要的问题.建立了一种利用广义Virasoro对称性的高维实现首先找到了一些(3+1)维Virasoro可积模型,并证明(3+1)维Virasoro可积模型均具有KacMoodyVirasoro对称代数.更进一步,利用WeissTaborCarnevale的奇性分析方法,证明了其中一个Virasoro可积模型也是Painlev啨可积的. 关键词： 广义Virasoro代数 Painlevé性质 (3+1)维可积模型     

利用Miura型不可逆变换得到高维可积模型

寻找高维可积模型(特别是3+1维可积模型)是非线性物理中的一个非常重要的问题.建立了一 种利用不可逆形变关系系统寻求高维可积模型的方法.不可逆形变既可以使可积模型成为不 可积模型,也可以使不可积模型成为可积模型.利用一种不可逆的Miura型形变关系和线性波 动方程,得到了一个非平庸的Painlevé可积的高维非线性模型. 关键词： 高维可织模型 不可逆形变 波动方程 Miura型变换     

具有阻尼项的非线性波动方程的相似约化

利用Clarkson和Kruskal引入的直接约化法,给出了具有阻尼项的非线性波动方程u_tt-2bu_xxt+αu_xxxx=β(uⁿ_x)x(α>0,β≠0,n≥2)三种类型的相似约化.从这些约化方程的Painlevé分析表明该方程在Ablowitz的猜测意义下是不可积的.此外还获得了该方程(n=2)的4种精确类孤波解. 关键词： 波动方程 相似约化 Painlevé分析 精确解     

一类非线性演化方程的新多级准确解

在Lamé方程和新的Lamé函数的基础上，应用小扰动方法和Jacobi椭圆函数展开法求解一类非线性演化方程（如mKdV方程，非线性Klein-Gordon方程Ⅱ等），获得多种新的多级准确解 .这些多级准确解对应着不同形式的周期波解.这些解在极限条件下可以退化为多种形式的孤 立波解，如带状孤立子、钟形孤立子等. 关键词： Jacobi椭圆函数 Lam函数 多级准确解 非线性演化方程 扰动方法     

具有物理背景的高维Painlevé可积模型

提出了一种求解任意维数非线性模型的“M?bious”变换下不变的渐进展开方法,并可同时获得许多新的与原模型有着相同维数的Painlevé可积模型.取(2+1)维KdV-Burgers(KdVB)方程和Kadomtsev-Petviashvili(KP)方程为具体例子,获得了一些新的具有Painlevé性质的高维“M?bious”变换下不变的方程及原模型的近似解.在某些特殊情况下,某些近似解可以成为精确解 关键词： 高维可积模型 “M?bious”不变 近似方法     

几类新的可积非线性色散项方程及其孤立波解

在不同系统参数下,通过Painlevé分析, 获得了广义修正Dullin-Gottwald-Holm方程中几类新的可积模型. 利用自动Backlund变换,得到了这几类可积模型的孤立波解. 关键词： Painlevé分析 广义修正Dullin-Gottwald-Holm方程 可积模型     

Gerdjikov-Ivanov方程的精确解

研究在量子场理论、弱非线性色散水波、非线性光学等领域中出现的Gerdjikov-Ivanov方程.对Gerdjikov-Ivanov方程的研究会导出具有高次非线性项的非线性数学物理方程.选取Liénard方程作为辅助常微分方程,借助于它并根据齐次平衡原则,求解了Gerdjikov-Ivanov方程,得到了该方程的包络孤立波解和包络正弦波解. 关键词： 齐次平衡原则 F展开法 Gerdjikov-Ivanov方程 包络孤立波解     

BBM方程和修正的BBM方程新的精确孤立波解

采用一种双曲函数假设和一类新的辅助常微分方程相结合的方法给出BBM方程和修正的BBM 方程新的精确孤立波解.这种方法也可用于寻找其他非线性发展方程新的孤立波解. 关键词： 辅助方程 双曲函数假设 孤立波解     

Boussinesq方程的Lax对、B?cklund变换、对称群变换和Riccati展开相容性

Boussinesq方程是流体力学等领域一个非常重要的方程.本文推导了Boussinesq方程的Lax对.借助于截断Painlevé展开,得到了Boussinesq方程的自B?cklund变换,以及Boussinesq方程和Schwarzian形式的Boussinesq方程之间的B?cklund变换.探讨了Boussinesq方程的非局域对称,研究了Boussinesq方程的单参数群变换和单参数子群不变解.运用Riccati展开法研究了Boussinesq方程,证明Boussinesq方程具有Riccati展开相容性,得到了Boussinesq方程的孤立波-椭圆余弦波解.     

On the Well-Posedness Problem and the Scattering Problem for the Dullin-Gottwald-Holm Equation

In this paper, we study the well-posedness of the Cauchy problem and the scattering problem for a new nonlinear dispersive shallow water wave equation (the so-called DGH equation) which was derived by Dullin, Gottwald and Holm. The issue of passing to the limit as the dispersive parameter tends to zero for the solution of the DGH equation is investigated, and the convergence of solutions to the DGH equation as ²0 is studied, and the scattering data of the scattering problem for the equation can be explicitly expressed; the new exact peaked solitary wave solutions are obtained in the DGH equation. After giving the condition of existing peakon in the DGH equation, it turns out to be nonlinearly stable for the peakon in the DGH equation.     

Optical Solitary Waves in Fourth-Order Dispersive Nonlinear Schr(o)dinger Equation with Self-steepening and Self-frequency Shift

By making use of the generalized sine-Gordon equation expansion method, we find cnoidal periodic wave solutions and fundamental bright and dark optical solitarywave solutions for the fourth-order dispersive and the quintic nonlinear Schrodinger equation with self-steepening, and self-frequency shift. Moreover, we discuss the formation conditions of the bright and dark solitary waves.     

Solitary waves and rogue waves in a plasma with nonthermal electrons featuring Tsallis distribution

In this Letter, we discuss the electron acoustic (EA) waves in plasmas, which consist of nonthermal hot electrons featuring the Tsallis distribution, and obtain the corresponding governing equation, that is, a nonlinear Schrödinger (NLS) equation. By means of Modulation Instability (MI) analysis of the EA waves, it is found that both electron acoustic solitary wave and rogue wave can exist in such plasmas. Basing on the Darboux transformation method, we derive the analytical expressions of nonlinear solutions of NLS equations, such as single/double solitary wave solutions and single/double rogue wave solutions. The existential regions and amplitude of solitary wave solutions and the rogue wave solutions are influenced by the nonextensive parameter q and nonthermal parameter α. Moreover, the interaction of solitary wave and how to postpone the excitation of rogue wave are also studied.     

Generalized and Improved （G＇/G）-Expansion Method Combined with Jacobi Elliptic Equation

In this article, we propose an alternative approach of the generalized and improved （G＇/G）-expansion method and build some new exact traveling wave solutions of three nonlinear evolution equations, namely the Boiti- Leon-Pempinelle equation, the Pochhammer-Chree equations and the Painleve integrable Burgers equation with free parameters. When the free parameters receive particular values, solitary wave solutions are constructed from the traveling waves. We use the Jacob/elliptic equation as an auxiliary equation in place of the second order linear equation. It is established that the proposed algorithm offers a further influential mathematical tool for constructing exact solutions of nonlinear evolution equations.     

A(2+1)-dimensional nonlinear model for Rossby waves in stratified fluids and its solitary solution

In this paper,we investigate a(2+1)-dimensional nonlinear equation model for Rossby waves in stratified fluids.We derive a forced Zakharov–Kuznetsov(ZK)–Burgers equation from the quasigeostrophic potential vorticity equation with dissipation and topography under the generalized beta effect,and by utilizing temporal and spatial multiple scale transform and the perturbation expansion method.Through the analysis of this model,it is found that the generalized beta effect and basic topography can induce nonlinear waves,and slowly varying topography is an external impact factor for Rossby waves.Additionally,the conservation laws for the mass and energy of solitary waves are analyzed.Eventually,the solitary wave solutions of the forced ZK–Burgers equation are obtained by the simplest equation method as well as the new modified ansatz method.Based on the solitary wave solutions obtained,we discuss the effects of dissipation and slowly varying topography on Rossby solitary waves.     

Stability of solitary wave trains in Hamiltonian wave systems

A class of Hamiltonian nonlinear wave equations possessing complex solitary waves with exponential decay is studied. It is shown that the interpulse interactions in a train of nearly identical solitary waves with large separations between the individual solitary waves are approximately described by a double Toda lattice system, with two variables at each lattice site. Under certain conditions, which are explicitly identified as Cauchy-Riemann equations, the two dynamical variables are real and imaginary parts of a single complex variable, leading to the complex Toda lattice equations, which is a discrete integrable dynamical system. This analysis generalizes to certain nonintegrable partial differential equations a recent result for the nonlinear Schr?dinger equation, and is important for the study of nonlinear communications channels in optical fibers. An example, the cubic-quintic nonlinear Schr?dinger equation, is worked out in detail to show that the theory can be carried through analytically. The theory is used to determine the stability of an infinite chain of nearly identical pulses separated by large time intervals. The entire theory is nonperturbative in the sense that the nonlinear wave equation need not be a weak perturbation of an integrable one.     

From bell-shaped solitary wave to W/M-shaped solitary wave solutions in an integrable nonlinear wave equation

The bifurcation theory of dynamical systems is applied to an integrable nonlinear wave equation. As a result, it is pointed out that the solitary waves of this equation evolve from bell-shaped solitary waves to W/M-shaped solitary waves when wave speed passes certain critical wave speed. Under different parameter conditions, all exact explicit parametric representations of solitary wave solutions are obtained.     

New Exact Travelling Wave Solutions for Zakharov-Kuznetsov Equation

In this paper, the nonlinear dispersive Zakharov- Kuznetsov equation is solved by using the generalized auxiliary equation method. As a result, new solitary pattern, solitary wave and singular solitary wave solutions are found.     

Abundant Multisoliton Structures of the Generalized Nizhnik-Novikov-Veselov Equation

Using the extended homogenous balance method, we obtainabundant exact solution structures ofa (2 1)dimensional integrable model, the generalized Nizhnik-Novikov-Veselov equation. By means of the leading order termanalysis, the nonlinear transformations of generalized Nizhnik-Novikov-Veselov equation are given first, and then somespecial types of single solitary wave solution and the multisoliton solutions are constructed.