Soliton Interactions of the Kadomtsev–Petviashvili Equation and Generation of Large-Amplitude Water Waves

We study the maximum wave amplitude produced by line-soliton interactions of the Kadomtsev–Petviashvili II (KPII) equation, and we discuss a mechanism of generation of large amplitude shallow water waves by multi-soliton interactions of KPII. We also describe a method to predict the possible maximum wave amplitude from asymptotic data. Finally, we report on numerical simulations of multi-soliton complexes of the KPII equation which verify the robustness of all types of soliton interactions and web-like structure.     

广义Davey-Stewartson方程的孤波解

在本文中,利用辅助的常微分方程来代替更一般的方程,扩张映射方法被呈现.通过这种方法,使用符号计算关于Davey-Stewartson方程的许多新的孤波解被构造.     

On the Blow-Up of Solutions of a Periodic Shallow Water Equation

Received April 28, 1999; accepted December 10, 1999     

变系数浅水波方程的精确解

均衡作用法给出了一种求非线性发展方程孤波解的有效方法.利用该方法,运用计算机符号计算,求出了变系数的一般浅水波方程的孤子解.     

正弦-Gordon方程n孤子解的一致性

正弦-Gordon方程是一种重要的非线性波动方程,其n孤子解具有Hirota表示与Wronski行列式表示形式,利用行列式的性质说明正弦-Gordon方程的这两种n孤子解的表示是一致的.     

Solutions of the Camassa-Holm Equation Near the Soliton

In this paper, solutions of the Camassa-Holm equation near the soliton Q is decomposed by pseudoconformal transformation as follows: λ^1/2(t)u(t, λ(t)y + x(t)) = Q(y) + ε(t, y), and the estimation formula with respect to ε(t, y) is obtained: |ε(t, y)| ≤ Ca₃Te^-θ|y|+ |λ^1/2(t)ε₀|. For the CH equation, we prove that the solution of the Cauchy problem and the soliton Q is sufficiently close as y → ∞, and the approximation degree of the solution an...     

广义随机KP方程的椭圆周期解

利用Hermite变换和Jacobi椭圆函数展开法研究(2+1)-维广义随机Kadomtsev-Petviashvili方程,并给出了它的随机椭圆周期解及随机孤立波解.     

广义浅水波方程新的行波解

通过利用新的G展开法,并借助Mathematica计算软件,研究了广义浅水波方程的精确解,获得了该方程的含有多个任意参数的新的显式行波解,分别为三角函数解、双曲函数解、有理函数解和指数函数解,扩大了该类方程的解的范围.     

The Hirota Method and Soliton Solutions to the Multidimensional Nonlinear Schrodinger Equation

Using the Hirota method, we obtain a 1-soliton solution to the (3+1)-dimensional nonlinear Schrodinger equation.     

浅水波方程和Klein-Gordon方程新的精确行波解

采用了一种新的方法来求解浅水波方程和Klein-Gordon的行波解.在该方法下,Klein-Gordon方程和浅水波方程都得到了其精确的周期孤立波解,从而该方法的有效性得到了验证.     

Interaction of Shallow Water Waves

In this paper, we consider the Riemann problem and interaction of elementary waves for a nonlinear hyperbolic system of conservation laws that arises in shallow water theory. This class of equations includes as a special case the equations of classical shallow water equations. We study the bore and dilatation waves and their properties, and show the existence and uniqueness of the solution to the Riemann problem. Towards the end, we discuss numerical results for different initial data along with all possible interactions of elementary waves. It is noticed that in contrast to the p -system, the Riemann problem is solvable for arbitrary initial data, and its solution does not contain vacuum state.     

Bilinearization and Soliton Solutions of the Supersymmetric Coupled KdV Equation

Theoretical and Mathematical Physics - We propose an N=1 supersymmetric generalization of the coupled Korteweg-de Vries (KdV) equation and use the Hirota superoperator to obtain a superfield...     

Transverse Spectral Stability of Small Periodic Traveling Waves for the KP Equation

The Kadomtsev–Petviashvili (KP) equation possesses a four‐parameter family of one‐dimensional periodic traveling waves. We study the spectral stability of the waves with small amplitude with respect to two‐dimensional perturbations which are either periodic in the direction of propagation, with the same period as the one‐dimensional traveling wave, or nonperiodic (localized or bounded). We focus on the so‐called KP‐I equation (positive dispersion case), for which we show that these periodic waves are unstable with respect to both types of perturbations. Finally, we briefly discuss the KP‐II equation, for which we show that these periodic waves are spectrally stable with respect to perturbations that are periodic in the direction of propagation, and have long wavelengths in the transverse direction.     

一类变系数Boussinesq型方程的Darboux变换及多孤子解

主要讨论在某种约束下,变系数Boussinesq型方程和变系数Broer-KaupKupershmidt方程之间的联系,构造变系数Broer-Kaup-Kupershmidt方程的另外一种Darboux变换,且应用Darboux变换得到变系数Boussinesq型方程的孤子解.     

KdV方程纯孤立子解的整体渐近性质

研究KdV方程纯孤立子解的整体渐近性质,证明了N-孤立子解一致收敛到N个单孤立子解的叠加.进而得到了N-孤立子解在L1-范数意义下的渐近结果,并借此阐述了纯孤立子解与一般速降解的差异.     

(3+1)-维广义随机KP方程的精确解

利用统一方式构造非线性偏微分方程行波解的广义Jacobi椭圆函数展开法和Hermite变换来研究(3+1)-维广义随机KP方程,给出了它的随机对偶周期和多孤子解.     

A Model Equation for Water Waves

Water waves are usually modeled as solutions of Laplace's equation in the fluid domain with appropriate nonlinear boundary conditions. Here we present a simple differential equation on the mean free surface, whose solutions behave like water waves.     

广义Pochhammer-Chree方程的新孤波解和余弦周期波解

本文利用假设待定法求出了具5阶非线性项的广义Pochhammer-Chree方程具双曲正割函数分式形式的2个新孤波解和6个余弦函数周期波解,并分别给出了它们的有界性条件.揭示了行波波速v的改变与钟状孤波解和余弦周期波解波形变化的相关性.