Travelling wave solutions for a second order wave equation of KdV type

The theory of planar dynamical systems is used to study the dynamical behaviours of travelling wave solutions of a nonlinear wave equations of KdV type.In different regions of the parametric space,sufficient conditions to guarantee the existence of solitary wave solutions,periodic wave solutions,kink and anti-kink wave solutions are given.All possible exact explicit parametric representations are obtained for these waves.     

Exact Explicit Peakon and Periodic Cusp Wave Solutions for Several Nonlinear Wave Equations

Using the method of dynamical systems for six nonlinear wave equations, the exact explicit parametric representations of the solitary cusp wave solutions and the periodic cusp wave solutions are given. These parametric representations follow that when travelling systems corresponding to these nonlinear wave equations has a singular straight line, under some parameter conditions, nonanalytic travelling wave solutions must appear. Dedicated to Professor Zhifen Zhang on the occasion of her 80th birthday     

Bifurcations of travelling wave solutions for Jaulent-Miodek equations

By using the theory of bifurcations of planar dynamic systems to the coupled Jaulent-Miodek equations,the existence of smooth solitary travelling wave solutions and uncountably infinite many smooth periodic travelling wave solutions is studied and the bifurcation parametric sets are shown.Under the given parametric conditions,all possible representations of explicit exact solitary wave solutions and periodic wave solutions are obtained.     

Exact traveling wave solutions to 2D-generalized Benney-Luke equation

By using the dynamical system method to study the 2D-generalized Benney- Luke equation, the existence of kink wave solutions and uncountably infinite many smooth periodic wave solutions is shown. Explicit exact parametric representations for solutions of kink wave, periodic wave and unbounded traveling wave are obtained.     

Exact traveling wave solutions for an integrable nonlinear evolution equation given by M. Wadati

By using the method of dynamical systems, the travelling wave solutions of for an integrable nonlinear evolution equation is studied. Exact explicit parametric representations of kink and anti-kink wave, periodic wave solutions and uncountably infinite many smooth solitary wave solutions are given.     

Bifurcations of travelling wave solutions for a coupled nonlinear wave system

Introduction FangShaomeiandGuoBoling[1]consideredthefollowingtimeperiodicproblemof dampedcouplednonlinearwaveequations:ut f(u)x-αuxx βuxxx 2vvx=G1(u,v) h1(x),vt-γvxx 2(uv)x g(v)x=G2(u,v) h2(x),(1)whereα,β,γareconstants,andγ>0,β≠0.Undertheperiodicboundaryconditions,the authorsobtainedtheuniqueexistenceofstrongsolutionsfortheabovesystem.InthispaperweshallconsiderbifurcationbehaviorofthetravellingwavesolutionsofEq.(1)inthecaseGi(u,v)≡0,hi(u,v)≡0(i=1,2).Letξ=x-ct,u=u(x-ct),where cis…     

Kink wave determined by parabola solution of a nonlinear ordinary differential equation

By finding a parabola solution connecting two equilibrium points of a planar dynamical system,the existence of the kink wave solution for 6 classes of nonlinear wave equations is shown.Some exact explicit parametric representations of kink wave solutions are given.Explicit parameter conditions to guarantee the existence of kink wave solutions are determined.     

Existence and breaking property of real loop-solutions of two nonlinear wave equations

Dynamical analysis has revealed that, for some nonlinear wave equations, loop- and inverted loop-soliton solutions are actually visual artifacts. The so-called loopsoliton solution consists of three solutions, and is not a real solution. This paper answers the question as to whether or not nonlinear wave equations exist for which a ＂real＂ loopsolution exists, and if so, what are the precise parametric representations of these loop traveling wave solutions.     

Bifurcations of traveling wave solutions and exact solutions to generalized Zakharov equation and Ginzburg-Landau equation

This paper studies the dynamic behaviors of some exact traveling wave solutions to the generalized Zakharov equation and the Ginzburg-Landau equation. The effects of the behaviors on the parameters of the systems are also studied by using a dynamical system method. Six exact explicit parametric representations of the traveling wave solutions to the two equations are given.     

WAVE INTERACTIONS IN SULICIU MODEL FOR DYNAMIC PHASE TRANSITIONS

Elementary waves in Suliciu model for dynamic phase transitions are obtained through traveling wave analysis.For any given initial data with two pieces of constant states,the Riemann solutions are constructed as a combination of elementary waves. When the initial profile contains three pieces of constant states,the solution may be constructed from the Riemann solutions,with each two adjacent states connected by elementary waves.A new Riemann problem forms when these two waves collide.Through the exploration of these Riemann problems,the outcome of wave interactions may be classified in a suitable parametric space.     

A class of coupled nonlinear Schr(o)dinger equations: Painlevé property,exact solutions,and application to atmospheric gravity waves

The Painlev(e) integrability and exact solutions to a coupled nonlinear Schrodinger (CNLS) equation applied in atmospheric dynamics are discussed. Some parametric restrictions of the CNLS equation are given to pass the Painleve test. Twenty periodic cnoidal wave solutions are obtained by applying the rational expansions of fundamental Jacobi elliptic functions. The exact solutions to the CNLS equation are used to explain the generation and propagation of atmospheric gravity waves.     

New groups of solutions to the Whitham-Broer-Kaup equation

The Whitham-Broer-Kaup model is widely used to study the tsunami waves. The classical Whitham-Broer-Kaup equations are re-investigated in detail by the generalized projective Riccati-equation method. 20 sets of solutions are obtained of which, to the best of the authors’ knowledge, some have not been reported in literature. Bifurcation analysis of the planar dynamical systems is then used to show different phase portraits of the traveling wave solutions under various parametric conditions.     

A class of coupled nonlinear Schrödinger equations: Painlevé property,exact solutions,and application to atmospheric gravity waves

The Painleve integrability and exact solutions to a coupled nonlinear Schrodinger （CNLS） equation applied in atmospheric dynamics are discussed. Some parametric restrictions of the CNLS equation are given to pass the Painleve test. Twenty periodic cnoidal wave solutions are obtained by applying the rational expansions of fundamental Jacobi elliptic functions. The exact solutions to the CNLS equation are used to explain the generation and propagation of atmospheric gravity waves.     

有限厚度砂床对波浪载荷的响应

本文采用水-土两相介质混合物的连续介质力学理论分析了有限厚度砂床对波浪载荷的响应。土床看作饱和孔隙弹性介质,用Biot理论来描述流动和变形的耦合效应,波浪用线性水波描述。在上述假设下得到了描述土床对波浪载荷响应的基本方程。用解沂-数值方法给出了土床中位移、应力和孔隙水压力的分布并将理沦计算结果与波浪-砂床模型试验的实验资料进行了比较。文章还对影响土床对波浪响应的参量特性进行了分析和讨论。根据参量研究、理论计算与实验资料比较的结果,可以得出结论:本文所提供的理论模型和计算方法能用来估算海床对波浪载荷的影响。     

The bifurcation and exact travelling wave solutions of (1+2)-dimensional nonlinear Schrödinger equation with dual-power law nonlinearity

By using the method of dynamical systems, this paper researches the bifurcation and the exact traveling wave solutions for a (1+2)-dimensional nonlinear Schrödinger equation with dual-power law nonlinearity. Exact parametric representations of all wave solutions are given.     

Dispersion Waves of Two Fluids in a Porous Medium

The dispersion of two fluids in a porous medium is analyzed as a wave process. The wave equations are derived, and for plane wave solutions a wave number versus frequency dispersion relation is obtained. Suitable choices for the saturation dependence of terms in the equations of motion and the dynamic pressure difference equation lead to physical solutions.     

Shape analysis and damped oscillatory solutions for a class of nonlinear wave equation with quintic term简

This paper aims at analyzing the shapes of the bounded traveling wave solu- tions for a class of nonlinear wave equation with a quintic term and obtaining its damped oscillatory solutions. The theory and method of planar dynamical systems are used to make a qualitative analysis to the planar dynamical system which the bounded traveling wave solutions of this equation correspond to. The shapes, existent number, and condi- tions are presented for all bounded traveling wave solutions. The bounded traveling wave solutions are obtained by the undetermined coefficients method according to their shapes, including exact expressions of bell and kink profile solitary wave solutions and approxi- mate expressions of damped oscillatory solutions. For the approximate damped oscillatory solution, using the homogenization principle, its error estimate is given by establishing the integral equation, which reflects the relation between the exact and approximate so- lutions. It can be seen that the error is infinitesimal decreasing in the exponential form.     

Traveling Wave Solutions for a Class of One-Dimensional Nonlinear Shallow Water Wave Models

In this paper we consider a class of one-dimensional nonlinear shallow water wave models that support weak solutions. We construct new traveling wave solutions for these models. Moreover, we show that these new traveling wave solutions are stable.     

Classification of Traveling Waves for a Class of Nonlinear Wave Equations

We classify the weak traveling wave solutions for a class of one-dimensional non-linear shallow water wave models. The equations are shown to admit smooth, peaked, and cusped solutions, as well as more exotic waves such as stumpons and composite waves. We also explain how some previously studied traveling wave solutions of the models fit into this classification.     

Generalized Darboux transformation and localized waves in coupled Hirota equations

In this paper, we construct a generalized Darboux transformation to the coupled Hirota equations with high-order nonlinear effects like the third dispersion, self-steepening and inelastic Raman scattering terms. As application, an N$N$th-order localized wave solution on the plane backgrounds with the same spectral parameter is derived through the direct iterative rule. In particular, some semi-rational, multi-parametric localized wave solutions are obtained: (1) vector generalization of the first- and the second-order rogue wave solutions; (2) interactional solutions between a dark–bright soliton and a rogue wave, two dark–bright solitons and a second-order rogue wave; (3) interactional solutions between a breather and a rogue wave, two breathers and a second-order rogue wave. The results further reveal the striking dynamic structures of localized waves in complex coupled systems.