受迫耗散Boussinesq方程的可积性质和孤波解的探讨

考虑到耗散效应和地形外力,Rossby波的振幅可由受迫耗散Boussinesq方程来描述.当包含这两项时,模型比较复杂,不具有Painleve性质.通过将模型双线性化,双线性方法是一个可寻找孤波解和B(a|¨)cklund变换的方法.通过截断的Painleve展开式,得到了将方程双线性化的合适的因变量变换.然后得到了受迫耗散Boussinesq方程的单孤波解和B(a|¨)cklund变换.     

Painleve Property and New Analytic Solutions for a Variable-Coefficient Kadomtsev--Petviashvili Equation with Symbolic Computation

A variable-coemcient Kadomtsev-Petviashvili equation is investigated. The Painlev6 analysis leads to its explicit Painlevd-integrable conditions. An auto-Backlund transformation and the bilinear form are presented via the truncated Painlev6 expansion and symbolic computation. Several families of new analytic solutions arepresented, including the soliton-like and periodic solutions.     

Baicklund Transformation and Multisoliton Solutions in Terms of Wronskian Determinant for （2~-l）-Dimensional Breaking Soliton Equations with Symbolic Computation

In this paper, two types of the （2＋1）-dimensional breaking soliton equations axe investigated, which describe the interactions of the Riemann waves with the long waves. With symbolic computation, the Hirota bilineax forms and Bgcklund transformations are derived for those two systems. Furthermore, multisoliton solutions in terms of the Wronskian determinant are constructed, which are verified through the direct substitution of the solutions into the bilineax equations. Via the Wronskian technique, it is proved that the Bgcklund transformations obtained are the ones between the （ N - 1）- and N-soliton solutions. Propagations and interactions of the kink-/bell-shaped solitons are presented. It is shown that the Riemann waves possess the solitonie properties, and maintain the amplitudes and velocities in the collisions only with some phase shifts.     

Existence of Formal Conservation Laws of a Variable-Coefficient Korteweg--de Vries Equation from Fluid Dynamics and Plasma Physics via Symbolic Computation

Employing the method which can be used to demonstrate the infinite conservation laws for the standard Kortewegde Vries （KdV） equation, we prove that the variable-coeFficient KdV equation under the Painlevé test condition also possesses the formal conservation laws.     

N-Soliton Solution in Wronskian Form for a Generalized Variable-Coefficient Korteweg--de Vries Equation

We concentrate on finding exact solutions for a generalized variable-coefficient Korteweg-de Vries equation of physically significance. The analytic N-soliton solution in Wronskian form for such a model is postulated and verified by direct substituting the solution into the bilinear form by virtue of the Wronskian technique. Additionally, the bilinear auto-Backlund transformation between the （ N - 1）- and N-soliton solutions is verified.     

Darboux Transformation and Soliton Solutions for the (2+1)-Dimensional Generalization of Shallow Water Wave Equation with Symbolic Computation

In this paper,the (2+1)-dimensional generalization of shallow water wave equation,which may be used to describe the propagation of ocean waves,is analytically investigated.With the aid of symbolic computation,we prove that the (2+1)-dimensional generalization of shallow water wave equation possesses the Painlev property under a certain condition,and its Lax pair is constructed by applying the singular manifold method.Based on the obtained Lax representation,the Darboux transformation (DT) is constructed.The first iterated solution,second iterated solution and a special N-soliton solution with an arbitrary function are derived with the resulting DT.Relevant properties are graphically illustrated,which might be helpful to understanding the propagation processes for ocean waves in shallow water.     

分离变量法在变系数(2+1)维方程求解中的应用

分离变量法是求解具有局域相干结构解的有效解析方法.考虑到传播介质的非均匀性和边界的不一致性,变系数(2+1)色散长波方程可以实际地描述宽广的河道或有限深的远海中非线性波的传播.解析研究了变系数(2+1)维色散长波方程.通过分离变量法,得到了该方程组的具有丰富结构的分离变量解.     

基于广义高阶非线性薛定谔方程深海内波传播的数值模拟

基于与实际海洋背景参数相关的广义高阶非线性薛定谔方程,首先讨论了不同的海洋环境参数对方程的非线性项和频散项的影响;然后通过有限差分算子给出了方程的二阶三层数值差分格式,并且分析了该差分格式的稳定性与精度阶;最后又通过得到的差分格式数值模拟了不同的海洋环境参数下深海内波的传播情况,结果显示:内波由深海向浅海的传播过程中,随着总水深的变化,发生了分裂现象,并且密度差之比越大,波的分裂速度越快.