Degasperis-Procesi方程的孤立尖波解

利用动力系统的定性分析理论对D egasperis-P rocesi方程的孤立尖波解进行了研究.给出了D e-gasperis-P rocesi方程对应行波系统的相图分支,利用相图获得了孤立尖波解和周期尖波解的解析表达式,通过数值模拟给出了部分解的图像.     

广义Gilson-Pickering方程的分支解

本文利用动力系统方法和奇行波方程理论研究广义Gilson-Pickering方程的动力学行为和行波解.利用软件画出了给定参数条件下系统的相图分支,得到了孤立波解、扭结波解和反扭结波解、不可数无穷多破缺波解、光滑周期波解和非光滑周期尖波解、尖孤子解的存在性.在β≠1,p=2时,对于广义Gilson-Pickering方程不同的参数条件下,给出了保证上述解存在的条件及参数表示.     

CH-r方程的尖波解

用动力系统的定性分析理论和数值模拟方法, 对CH-r方程的尖波解进行研究, 获得了孤立尖波和周期尖波的解析表达式, 揭示了这两种尖波解之间的一些关系, 还指出了产生尖点的原因.     

微结构固体中孤立波的演变及非光滑孤立波

给出了包含宏观应变和微形变的全部二次项以及宏观应变三次项的一种新的自由能函数.利用新自由能函数并根据Mindlin微结构理论，建立了描述微结构固体中纵波传播的一种新模型.利用近来发展的奇行波系统的动力系统理论，分析了系统的所有相图分支，并给出了周期波解、孤立波解、准孤立尖波解、孤立尖波解以及紧孤立波解.孤立尖波解和紧孤立波解的得到，有效地证明了在一定条件下，微结构固体中可以形成和存在孤立尖波和紧孤立波等非光滑孤立波.此结果进一步推广了微结构固体中只存在光滑孤立波的已有结论.     

Fokas方程的显示孤立尖波解和周期尖波解

利用一个独立变换和动力系统方法对Fokas方程:u_(tx)=(1+v(?)■_x~2)sin(u),x∈R,t0进行研究.在对该方程所对应的平面动力系统进行定性分析的基础上,得到了该方程所有可能的显式孤立尖波解和周期尖波解.     

广义Camassa-Holm方程的行波解

运用平面动力系统理论和方法给出了广义Camassa-Holm方程在各种参数条件下的相图与分支,分析了奇线对其行波解的影响,获得了广义Camassa-Holm方程光滑、非光滑孤立波解和周期波解的存在性及个数,求出了它的两组新周期尖波解的显式表达式.     

一类5阶KdV方程的孤立波解

近年来,关于3阶KdV方程的孤立波解得到迅速发展,而对于5阶KdV方程的孤立波解文献报道较少.本文主要采用Sine-Cosine展开法得到了一类5阶KdV方程的孤立波解;然后利用Matlab计算软件,获得了孤立波解的图形,其结果展示了孤立子与系数之间的相互关系;最后,应用所得的结果分别得到了Lax方程, SK方程, CDG方程的孤立波解.     

非线性波方程的精确孤立波解

立了一种求解非线性波方程精确孤立波解的双曲函数方法,并在计算机代数系统上加以实现,推导出了一大批非线性波方程的精确孤立波解.方法的基本原理是利用非线性波方程孤立波解的局部性特点,将方程的孤立波解表示为双曲函数的多项式,从而将非线性波方程的求解问题转化为非线性代数方程组的求解问题.利用吴消元法或Gröbner基方法在计算机代数系统上求解非线性代数方程组, 最终获得非线性波方程的精确孤立波解,其中有很多新的精确孤立波解.     

Degasperis-Procesi方程的类孤子新解

利用辅助方程与函数变换相结合的方法,构造了Degasperis-Procesi(D-P)方程的无穷序列类孤子新解.首先,通过两种函数变换,把D-P方程化为常微分方程组.然后,利用常微分方程组的首次积分,把D-P方程的求解问题化为几种常微分方程的求解问题.最后,利用几种常微分方程的Bcklund变换等相关结论,构造了D-P方程的无穷序列类孤子新解.这里包括由Riemannθ函数、Jacobi椭圆函数、双曲函数、三角函数和有理函数组成的无穷序列光滑孤立子解、尖峰孤立子解和紧孤立子解.     

浅水中度振幅孤立波解的分支

利用平面动力系统分支方法研究浅水中度振幅方程的定性行为和孤立波解.给出了系统在不同参数条件下的相图.获得了光滑孤立波、cuspon解和周期波解的隐式表达式.对方程的光滑孤立波解、cuspon解和周期波解进行了数值模拟.获得的结果完善了相关文献已有的结果.     

Traveling wave solutions of the Degasperis-Procesi equation

We classify all weak traveling wave solutions of the Degasperis-Procesi equation. In addition to smooth and peaked solutions, the equation is shown to admit more exotic traveling waves such as cuspons, stumpons, and composite waves.     

Solitons solutions for some nonlinear evolution equations

In this paper, we establish new solitary wave solutions to the modified Kawahara equation by the sine-cosine method. Moreover, the periodic solutions and bell-shaped solitons solutions to the generalized fifth-order KdV equation are obtained. The tanh method is used to handle the double sine-Gordon equation and the double sinh-Gordon equation. Families of exact travelling wave solutions are formally derived. The rational triangle sine-cosine method is introduced and to be constructed complex solutions to the modified Degasperis-Procesi (DP) equation and the modified Camassa-Holm (CH) equation.     

Single peak solitary wave solutions for the generalized Camassa–Holm equation

This Letter presents all possible smooth, peaked and cusped solitary wave solutions for the generalized Camassa–Holm equation under the inhomogeneous boundary condition.The parametric conditions of existence of the smooth, peaked and cusped solitary wave solutions are given by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for smooth, peaked and cusped solitary wave solutions of the generalized Camassa–Holm equation.     

Single peak solitary wave solutions for the Fornberg–Whitham equation

The qualitative theory of differential equations is applied to the Fornberg–Whitham equation. Smooth, peaked and cusped solitary wave solutions of the Fornberg–Whitham equation under inhomogeneous boundary condition are obtained. The conditions of existence of the smooth, peaked and cusped solitary wave solutions are given by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for smooth, peaked and cusped solitary wave solutions of the Fornberg–Whitham equation. The results presented in this article extend and improve the previous results.     

New exact solutions for mCH and mDP equations by auxiliary equation method

With the aid of symbolic computation, auxiliary equation method is introduced to investigate modified forms of Camassa-Holm and Degasperis-Procesi equations. A series of new exact traveling wave solutions, including smooth solitary wave solution, peakons, singular solution, periodic wave solution, Jacobi elliptic solution, are obtained in general form. These new exact solutions will enrich previous results and help us further understand the physical structures of these two nonlinear equations.     

PEAKED WAVE SOLUTIONS TO A GENERALIZED HYPERELASTIC-ROD WAVE EQUATION

We study peaked wave solutions of a generalized Hyperelastic-rod wave equation describing waves in compressible hyperelastic-rods by using the bifurcation theory of planar dynamical systems and numerical simulation method. The existence domain of the peaked solitary waves are found. The analytic expressions of peaked solitary wave solutions are obtained. Our numerical simulation and qualitative results are identical.     

Peaked wave solutions of CH-r equation

In this paper the qualitative analysis methods of planar autonomous systems and numerica simu-lation are used to investigate the peaked wave solutions of CH-r equation. Some explicit expressions of peakedsolitary wave solutions and peaked periodic wave solutions are obtained,and some of their relationships arerevealed.Why peaked points are generated is discussed.     

Stability of Solitary Waves and Global Existence of a Generalized Two-Component Camassa–Holm System

We study here the existence of solitary wave solutions of a generalized two-component Camassa–Holm system. In addition to those smooth solitary-wave solutions, we show that there are solitary waves with singularities: peaked and cusped solitary waves. We also demonstrate that all smooth solitary waves are orbitally stable in the energy space. We finally give a sufficient condition for global strong solutions to the equation in some special case.     

Factorization technique and new exact solutions for the modified Camassa-Holm and Degasperis-Procesi equations

In this paper, the recent factorization technique is applied to the modified Camassa-Holm and Degasperis-Procesi equations and two first-order ordinary differential equations are obtained, respectively. Subsequently, some new exact solitary wave solutions for the two equations are proposed. The figures for the bell-type and peakon-type solutions of the modified Camassa-Holm are plotted to describe the properties of the solutions.