Local existence of strong solutions to the generalized Boussinesq equations

This paper deals with the local existence and uniqueness of strong solutions for the generalized Boussinesq equations with fractional dissipation. As a corollary, we establish some regularity criteria to guarantee smoothness of solutions.     

非线性自治振动系统同宿解的广义双曲函数摄动法

提出广义的双曲函数摄动法,用于求解强非线性自治振子的同宿解,克服一般摄动步骤中派生方程须存在显式精确同宿解的限制．以广义双曲函数作为摄动步骤的基本函数,拓展了基于双曲函数的摄动法的适用范围．对同时含2,3次和含4次强非线性项的系统进行求解分析,验证了方法的有效性和精度．     

Exact Solitary-wave Solutions and Periodic Wave Solutions for Generalized Modified Boussinesq Equation and the Effect of Wave Velocity on Wave Shape

By means of the undetermined assumption method, we obtain some new exact solitary-wave solutions with hyperbolic secant function fractional form and periodic wave solutions with cosine function form for the generalized modified Boussinesq equation. We also discuss the boundedness of these solutions. More over, we study the correlative characteristic of the solitary-wave solutions and the periodic wave solutions along with the travelling wave velocity＇s variation.     

一类时滞Duffing微分方程同宿解的存在性

利用重合度理论和一些分析技巧,得到一类二阶时滞Duffing微分方程的2kT周期解,通过对该微分方程的一系列周期解取极限获得同宿解的存在性.同时,β(t)是可变号的.     

Solitary wave solutions to the generalized coupled mKdV equation with multi-component

Using the Darboux matrix method, the multi-solitary wave solutions of the generalized coupled mKdV equation with multi-component are obtained. The obtained solution formulas provide us with a comprehensive approach to construct exact solutions for the generalized coupled mKdV equation by some basic solutions of the Boiti and Tu spectral problem.     

Existence and scattering of small solutions to a Boussinesq type equation of sixth order

In this paper, we consider the existence and uniqueness of the global small solution as well as the small data scattering result to the Cauchy problem for a Boussinesq type equation of sixth order with the nonlinear term f(u) behaving as as u→0 in . The main method and techniques used in our paper are the Littlewood-Paley dyadic decomposition, the stationary phase estimate and some properties of Bessel function.     

New exact Jacobi elliptic functions solutions for the generalized coupled Hirota-Satsuma KdV system

In this paper, an generalized Jacobi elliptic functions expansion method with computerized symbolic computation is used for constructing more new exact Jacobi elliptic functions solutions of the generalized coupled Hirota-Satsuma KdV system. As a result, eight families of new doubly periodic solutions are obtained by using this method, some of these solutions are degenerated to solitary wave solutions and triangular functions solutions in the limit cases when the modulus of the Jacobi elliptic functions m → 1 or 0, which shows that the applied method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.     

Existence of positive ground state solutions for the coupled Choquard system with potential

In this paper, we study the following coupled Choquard system in ${\mathbb{R}}^N$: where $\alpha \in (0,N)$ and , in which $2_{\ast }^{\alpha }$ denotes $\frac{N+\alpha }{N-2}$ if $N\ge 3$ and $2_{\ast }^{\alpha }:=\infty$ if $N=1,2$. The function $I_{\alpha }$ is a Riesz potential. By using Nehari manifold method, we obtain the existence of a positive ground state solution in the case of bounded potential and periodic potential, respectively. In particular, the nonlinear term includes the well-studied case $p=q$ and $u(x)=v(x)$, and the less-studied case $p\ne q$ and $u(x)\ne v(x)$. Moreover, it seems to be the first existence result for the case $p\ne q$.     

Sufficient close-to-necessary conditions for the blowup of solutions to a strongly nonlinear generalized Boussinesq equation

An initial boundary value problem for the generalized Boussinesq equation with allowance for linear dissipation and free electron sources is considered. The strong generalized time-local solvability of the problem is proved. Sufficient conditions are obtained for the blowup of the solution and for time-global solvability. Two-sided estimates of the blowup time are derived.     

Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth systems with a cusp

For a piecewise analytical Hamiltonian system with a cusp on a switch line, which has a family of periodic orbits near a generalized homoclinic loop, we study the maximum number of limit cycles bifurcating from the periodic orbits. For doing so, we first obtain the asymptotic expressions of the Melnikov functions near the loop. Finally we present two examples illustrating applications of the main results.     

Existence and blow-up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation

We consider the existence, both locally and globally in time, and the blow-up of solutions for the Cauchy problem of the generalized damped multidimensional Boussinesq equation.     

On the Cauchy problem for a generalized Boussinesq equation

In this paper, we consider the Cauchy problem for a generalized Boussinesq equation. We show that, under suitable conditions, a global solution for the initial value problem exists. In addition, we derive the sufficient conditions for the blow-up of the solution to the problem.     

Solitonic solutions for a variable-coefficient variant Boussinesq system in the long gravity waves

Variable-coefficient variant Boussinesq (VCVB) system is able to describe the nonlinear and dispersive long gravity waves traveling in two horizontal directions with varying depth. In this paper, with symbolic computation, a Lax pair associated with the VCVB system under some constraints for variable coefficients is derived, and based on the Lax pair, two sorts of basic Darboux transformations are presented. By applying the Darboux transformations, some solitonic solutions are obtained, with the relevant constraints given in the text. In addition, the VCVB system is transformed to a variable-coefficient Broer-Kaup system. Solitonic solutions and procedure of getting them could be helpful to solve the nonlinear and dispersive problems in fluid dynamics.     

CTE method and exact solutions for modified Boussinesq system

In this paper, the truncated Painlevé analysis and the consistent tanh expansion method are developed for the modified Boussinesq system, and new exact solutions such as the single‐soliton, the two‐soliton, the rational solutions, and the explicit interaction solutions among a soliton and the cnoidal periodic waves are obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

Global existence and decay of solutions for the generalized bad Boussinesq equation

In this paper,we consider the global existence of solutions for the Cauchy problem of the generalized sixth order bad Boussinesq equation.Moreover,we show that the supremum norm of the solution decays algebraically to zero as(1+t)(1/7)when t approaches to infnity,provided the initial data are sufciently small and regular.