A stable finite-difference scheme for the Boussinesq equation

Summary  An explicit discretization scheme for the Boussinesq equation is developed and its (pseudo-) convergence and stability are investigated. This scheme is used to study the soliton properties in the model described by the Boussinesq equation. The emergence of soliton excitations out of prescribed multisoliton and nonsoliton initial conditions is also shown numerically. This work was supported in part by the NSF through the Grant Nr. HRD-9450342. One of us (CAC) wishes to thank the MURST for the grant awarded for the cohoperation between the Politecnico di Torino and the University of Puerto Rico.     

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

Using the extended homogenous balance method, we obtain abundant exact solution structures of a (2+1)-dimensional integrable model, the generalized Nizhnik-Novikov-Veselov equation. By means of the leading order term analysis, the nonlinear transformations of generalized Nizhnik-Novikov-Veselov equation are given first, and then some special types of single solitary wave solution and the multisoliton solutions are constructed.     

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.     

Exact Soliton Solutions for DNLS Equation by the Method of Riemann Problem with Zeros

With the procedure to solve explicitly the equations of the Riemann matrix with poles, multisoliton solutions to the DNLS equation are found formally. A single soliton solution is given explicitly and the asymptotic behaviors of multisoliton solutions are discussed.     

Mixed solutions of the sine-Gordon equation

The aim of this note is to present an explicit formula for the mixed solution of the sine-Gordon equation, i.e. the solution describing a multisoliton process on a background of multiphase quasi-periodic process.     

Some Special Types of Solitary Wave Solutions for (3+1)-Dimensional Jimbo-Miwa Equation

Using the extended homogeneous balance method, we find some special types of single solitary wave solution and new types of the multisoliton solutions of the (3+1)-dimensional Jimbo-Miwa equation.     

Study on an extended Boussinesq equation

An extended Boussinesq equation that models weakly nonlinear and weakly dispersive waves on a uniform layer of water is studied in this paper. The results show that the equation is not Painlev\'e-integrable in general. Some particular exact travelling wave solutions are obtained by using a function expansion method. An approximate solitary wave solution with physical significance is obtained by using a perturbation method. We find that the extended Boussinesq equation with a depth parameter of $1/\sqrt 2$ is able to match the Laitone's (1960) second order solitary wave solution of the Euler equations.     

Abundant Multisoliton Structure of the (3+1)-Dimensional Nizhnik-Novikov-Veselov Equation

Using the extended homogeneous balance method, we obtained abundant exact solution structures of the (3+1)-dimensional Nizhnik-Novikov-Veselov (NNV) equation. By means of the leading order term analysis, the nonlinear transformations of the (3+1)-dimensional NNV equation are given first, and then some special types of single solitary wave solution and the multisoliton solutions are constructed.     

A Variational Iteration Solving Method for a Class of Generalized Boussinesq Equations

We study a generalized nonlinear Boussinesq equation by introducing a proper functional and constructing the variational iteration sequence with suitable initial approximation. The approximate solution is obtained for the solitary wave of the Boussinesq equation with the variational iteration method.     

A New Multisoliton Solution to (2+1)-Dimensional KdV Equation

A new multisoliton solution to the (2+1)-dimensional KdV equation is obtained by means of the truncated Painleve expansion method and a direct ansatz technique. This new exact solution is periodic in the propagating direction x and exponentially decaying in y and thus it is called periodic solitons. A typical spatial structure of it is illustrated by the figures.     

A simple and direct method for generating travelling wave solutions for nonlinear equations

We propose a simple and direct method for generating travelling wave solutions for nonlinear integrable equations. We illustrate how nontrivial solutions for the KdV, the mKdV and the Boussinesq equations can be obtained from simple solutions of linear equations. We describe how using this method, a soliton solution of the KdV equation can yield soliton solutions for the mKdV as well as the Boussinesq equations. Similarly, starting with cnoidal solutions of the KdV equation, we can obtain the corresponding solutions for the mKdV as well as the Boussinesq equations. Simple solutions of linear equations can also lead to cnoidal solutions of nonlinear systems. Finally, we propose and solve some new families of KdV equations and show how soliton solutions are also obtained for the higher order equations of the KdV hierarchy using this method.     

Abundant Multisoliton Structure of (3+1)-Dimensional Breaking Soliton Equation

Using the extended homogeneous balance method, we obtained abundant exact solution structures of the (3 1 )-dimensional breaking soliton equation. By means of the leading order term analysis, the nonlinear transformations of the (3t1)-dimensional breaking soliton equation are given first, and then some special types of single solitary wave solutions and the multisoliton solutions are constructed.     

Dromion and other exact solutions of (2+1)-dimensional higher-order Broer-Kaup system

Using extended homogenous balance method, we obtain Bäcklund transformation (BT) and a linear partial differential equation of higher-order Broer-Kaup (HBK) system. As a result, multisoliton and single soliton and other exact solutions of (2+1)-dimensional HBK system are given. By analyzing single soliton solution, we get some dromion solutions.     

Linear stability of multiple internal solitary waves in fluids of great depth

The linear stability of the multiple solitary wave solution of the Benjamin-Ono (BO) equation is studied analytically. By establishing the completeness relation for the eigenfunctions of the BO equation linearized about multisoliton solutions, we solve the initial value problem for this system. We find that the wave under consideration is stable against infinitesimal perturbations.     

Abundant multisoliton structures of the Konopelchenko-Dubrovsky equation

Using the homogenous balance method, the nonlinear transformations of the (2+1)-dimensional integrable Konopelchenko-Dubrovsky equation are given, and then some new special types of single solitary wave solution and the multisoliton solutions are constructed. The project is supported by the Natural Science Foundation of Shandong Province in China and the Natural Science Foundation of Liaocheng University.     

Darboux Transformation for a Negative Order AKNS Equation

Using a quasideterminant Darboux matrix, we compute soliton solutions of a negative order AKNS (AKNS($-$1)) equation. Darboux transformation (DT) is defined on the solutions to the Lax pair and the AKNS($-$1) equation. By iterated DT to K-times, we obtain multisoliton solutions. It has been shown that multisoliton solutions can be expressed in terms of quasideterminants and shown to be related with the dressed solutions as obtained by dressing method.     

Invariant solutions and conservation laws of the generalized Kaup–Boussinesq equation

The generalized Kaup–Boussinesq equation is a model which is used to describe the water wave. In this paper, Lie group analysis method is used to perform detailed analysis on the generalized Kaup–Boussinesq equation. Some invariant solutions are obtained under the transformation groups. The conservation laws of the generalized Kaup–Boussinesq equation are constructed using two methods: multiplier method and Ibragimov theorem.     

A nonlocal Boussinesq equation: Multiple-soliton solutions and symmetry analysis

A nonlocal Boussinesq equation is deduced from the local one by using consistent correlated bang method. To study various exact solutions of the nonlocal Boussinesq equation, it is converted into two local equations which contain the local Boussinesq equation. From the N-soliton solutions of the local Boussinesq equation, the N-soliton solutions of the nonlocal Boussinesq equation are obtained, among which the (N=2,3,4)-soliton solutions are analyzed with graphs. Some periodic and traveling solutions of the nonlocal Boussinesq equation are derived directly from the known solutions of the local Boussinesq equation. Symmetry reduction solutions of the nonlocal Boussinesq equation are also obtained by using the classical Lie symmetry method.     

BOUSSINESQ方程初边值问题的孤立波解

本文给出了Boussinesq方程的一种差分格式并对其初边值问题进行了数值模拟。数值证明了在一些边界条件下Boussinesq方程孤立波解的存在。     

(2+1)维Boussinesq方程的新的周期解

利用Hirota方法及Riemann theta函数得到了(2+1)维Boussinesq方程的新的周期解.在极限情况下,该周期解退化为孤子解. 关键词： Hirota方法 Riemann theta 函数 (2+1)维Boussinesq方程 周期解