变系数KdV方程组的精确解

将Jacobi椭圆正弦函数展开法与Jacobi椭圆余弦函数展开法引入到变系数KdV方程组的求解中,得到了三组类周期波解．这些解析解在一定条件下退化为类孤波解．     

Lax形式的5阶KdV方程的两类尖孤立波解

Lax形式的5阶KdV方程的尖孤波解尚未见有文献报道.本文首次给出Lax形式的5阶KdV方程的两类尖孤波解.这两类孤波解都有尖峰或倒尖峰,且满足Rankine-Hugoniot条件和熵条件,是方程的物理解.     

粘弹性阻尼对弹性杆内MKdV应变孤波运动的影响

本文用连续谱的微扰论方法,详细地研究了非线性弹性杆内MKdV纵向应变孤波在粘弹性阻尼作用下的演变行为,并将其结果与KdV应变孤波的演变行为进行了比较。     

组合BBM-Burgers方程孤波解的稳定性

讨论了组合BBM-Burgers方程的孤立波解在Liapunov意义下的稳定性,证明了孤波解在初始微扰满足一定条件时具有条件稳定性.     

Burgers型方程初值问题解的稳定性及孤波解

本文证明Burgers型方程初值问题的解在L₂意义上的稳定性、唯一性;给出该方程存在扭状孤波解的条件及求扭状孤波解的方法。并指出Burgers型方程不存在钟状孤波解。     

Lamé函数和非线性演化方程的新多级准确解

基于Lamé方程和新的Lamé函数,应用摄动方法和Jacobi椭圆函数展开法求解非线性演化方程,获得多种新的多级准确解.这些解在极限条件下可以退化为各种形武的孤波解.     

求解非线性方程的双函数法

基于齐次平衡法和李志斌的tanh函数法,得到简单有效的求解非线性发展方程的双函数法,这种方法利用非线性发展方程孤立波的局部性特点,把非线性方程的孤波解表示为函数f和g的多项式,并用这种方法求出了非线性波理论中的基本模型KdV方程的多组孤波解。     

含任意次正幂项的广义五阶KdV方程的精确解

利用齐次平衡原理,通过引进含非线性辅助微分方程（sub-ODE）,获得了含任意次正幂项的广义五阶KdV方程的精确解,包括钟状孤波解,扭状孤波解和三角函数表示的周期波解.所得精确解与前人用其它方法所获得一致,并包含了以往文献未提供的部分解,扩充并完善了以往文献的相关结果.     

耦合Schrdinger和KdV方程组孤波解的存在性

主要研究了耦合的非线性Schrdinger和KdV方程孤波解的存在性.文章利用集中紧性原理找到预紧性的极小化序列,通过平移的方式来寻找方程组对应泛函在H~1(R)的极小值函数,从而得到原方程非平凡解的存在性.     

非线性波方程新的显式精确解

利用Darboux和一个可化为标准Bernoulli方程的4阶常微分方程,统一地处理了三个著名方程KdV方程,Kadomtsev-Petviashvili(KP)方程和Hirota-Satsuma(HS)方程的求解问题.给出了这些方程一批新的具有更为丰富形式的精确解,其中包括孤波解和行波解.     

A modified tanh–coth method for solving the KdV and the KdV–Burgers’ equations

In this work we use a modified tanh–coth method to solve the Korteweg-de Vries and Korteweg-de Vries–Burgers’ equations. The main idea is to take full advantage of the Riccati equation that the tanh-function satisfies. New multiple travelling wave solutions are obtained for the Korteweg-de Vries and Korteweg-de Vries–Burgers’ equations.     

受迫广义KdV-Burgers方程行波解的存在唯一性

本文研究具有受迫性的广义二维KdV-Burgers方程的周期行波解,为了获得周期行波解的存在唯一性定理,使唤用特定系数法和Schauder不动点定理获得了受迫广义KdV-Burgers方程周期行波解存在唯一性的条件.并获得了周期行波解的一些先验估计式.     

Initial Value Problem for a Generalized Korteweg－de Vries Equation with Singular Integral－Differential Terms

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ASYMPTOTIC PROPERTY FOR THE SOLUTION TO THE GENERALIZED KORTEWEG－DE VRIES EQUATION

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Solitary wave and shock wave solutions to a second order wave equation of Korteweg-de Vries type

This paper obtains the solitary wave as well as the shock wave solutions to the second order wave equation of Korteweg-de Vries type that was first proposed in 2002. The ansatz method is used to retrieve these solutions. The domain restrictions as well as the parameter regimes are all identified in the process of obtaining the solution.     

Interactions between exotic multi-valued solitons of the (2+1)-dimensional Korteweg-de Vries equation describing shallow water wave

Korteweg-de Vries equation governs the weakly nonlinear long wave whose phase speed reaches a simple maximum of wave with the infinite length in shallow water wave. The exponential-form variable separation solution of (2+1)-dimensional Kortweg-de Vries equation is found via the two-function method, and this solution covers many special combined solutions including sinh-cosh,sin-cos,sech-tanh,csch-coth,sec-tan and csc-cot solutions. From the exponential-form solution with choosing suitable functions, inelastic interactions between special multi-valued solitons with two loops such as anti-bell-shaped, anti-peak-shaped semifoldons and anti-foldon are graphically and analytically studied. By the asymptotic analysis, phase shift and its difference during interactions between multi-valued solitons are analytically given.     

The Korteweg-de Vries equation in a cylindrical pipe

A mathematical model of nonlinear wave propagation in a pipeline is constructed. The Korteweg-de Vries equation is derived by determining asymptotic solutions and changing variables. A particular solution to the model equations is found that has the fluid velocity function in the form of a solitary wave. Thus, the class of nonlinear fluid dynamics problems described by the KdV equation is expanded.     

Weak and Strong Interactions between Internal Solitary Waves

The interaction of weakly nonlinear long internal gravity waves is studied. Weak interactions occur when the wave phase speeds are unequal; this case includes that of a head-on collision. It is shown that each wave satisfies a Korteweg-de Vries equation, and the main effect of the interaction is described by a phase shift. Strong interactions occur when the wave phase speeds are nearly equal although the waves belong to different modes. This case is described by a pair of coupled Korteweg-de Vries equations, for which some preliminary numerical results are presented.     

On “new travelling wave solutions” of the KdV and the KdV–Burgers equations

The Korteweg-de Vries and the Korteweg-de Vries–Burgers equations are considered. Using the travelling wave the general solutions of these equations are presented. “New travelling wave solutions” of the KdV and the KdV–Burgers equations by Wazzan [Wazzan L. Commun Nonlinear Sci Numer Simulat 2009;14:443–50] are analyzed. We demonstrate that all his solutions are not new and are transformed to known solutions.     

Topological 1-soliton solution of the generalized KdV equation with generalized evolution

In this paper, the topological 1-soliton solution to the generalized Korteweg-de Vries equation with generalized evolution will be obtained. The solitary wave ansatz methods will be applied to obtain the solutions. In the process, the constraints relations between the soliton parameters will also be determined.