Different physical structures of solutions for a generalized Boussinesq wave equation

A technique based on the reduction of order for solving differential equations is employed to investigate a generalized nonlinear Boussinesq wave equation. The compacton solutions, solitons, solitary pattern solutions, periodic solutions and algebraic travelling wave solutions for the equation are expressed analytically under several circumstances. The qualitative change in the physical structures of the solutions is highlighted.     

一类耦合非线性波方程的行波解分支

利用动力系统的Hopf分支理论来研究耦合非线性波方程周期行波解的存在性和稳定性．应用行波法把一类耦合非线性波方程转换为三维动力系统来研究,从而给在不同的参数条件下给出了周期解存在和稳定性的充分条件．     

On the linear stability of simple and semi-simple periodic waves for a system of cubic Klein–Gordon equations

We study the linear stability of traveling wave solutions for the nonlinear wave equation and coupled nonlinear wave equations. It is shown that periodic waves of the dnoidal type are spectrally unstable with respect to co-periodic perturbations. Our arguments rely on a careful spectral analysis of various self-adjoint operators, both scalar and matrix and on instability index count theory for Hamiltonian systems.     

Instability of cnoidal‐peak for the NLS‐δ equation

We study the existence and stability of standing waves for the periodic cubic nonlinear Schrödinger equation with a point defect determined by the periodic Dirac distribution at the origin. We show that this model admits a smooth curve of periodic‐peak standing wave solutions with a profile determined by the Jacobi elliptic function of cnoidal type. Via a perturbation method and continuation argument, we obtain that in the repulsive defect, the cnoidal‐peak standing wave solutions are unstable in $H^1_{per}$ with respect to perturbations which have the same period as the wave itself. Global well‐posedness is verified for the Cauchy problem in $H^1_{per}$ .     

Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka–Volterra competition system with diffusion

We study the existence, uniqueness, and asymptotic stability of time periodic traveling wave solutions to a periodic diffusive Lotka–Volterra competition system. Under certain conditions, we prove that there exists a maximal wave speed c^? such that for each wave speed c?c^?, there is a time periodic traveling wave connecting two semi-trivial periodic solutions of the corresponding kinetic system. It is shown that such a traveling wave is unique modulo translation and is monotone with respect to its co-moving frame coordinate. We also show that the traveling wave solutions with wave speed c<c^? are asymptotically stable in certain sense. In addition, we establish the nonexistence of time periodic traveling waves for nonzero speed c>c^?.     

Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis

In this paper, we establish the existence and the nonlinear stability of traveling wave solutions to a system of conservation laws which is transformed, by a change of variable, from the well-known Keller-Segel model describing cell (bacteria) movement toward the concentration gradient of the chemical that is consumed by the cells. We prove the existence of traveling fronts by the phase plane analysis and show the asymptotic nonlinear stability of traveling wave solutions without the smallness assumption on the wave strengths by the method of energy estimates.     

Global entropy solutions to a class of quasi-linear wave equations with large time-oscillating sources

This research explores the Cauchy problem for a class of quasi-linear wave equations with time dependent sources. It can be transformed into the Cauchy problem of hyperbolic integro-differential systems of nonlinear balance laws. We introduce the generalized Glimm scheme in new version and study its stability which is proved by Glimm-type interaction estimates in a dissipativity assumption. The generalized solutions to the perturbed Riemann problems, the building blocks of generalized Glimm scheme, are constructed by Riemann problem method modeled on the source free equations. The global existence for the Lipschitz continuous solutions and weak solutions to the systems is established by the consistency of scheme and the weak convergence of source. Finally, the weak solutions are also the entropy solutions which satisfy the entropy inequality.     

NEW PERIODIC SOLUTIONS OF ITO＇S 5th-ORDER mKdV EQUATION AND ITO＇S 7th-ORDER mKdV EQUATION

Based on the modified Jocobi elliptic function expansion method and the modified extended tanh-function method, a new algebraic method is presented to obtain multiple travelling wave solutions for nonlinear wave equations. By using the method ,Ito‘s 5th-order and 7th-order mKdV equations are studied in detail and more new exact Jocobi elliptic function periodic solutions are found. With modulus m→1 or m→0, these solutions degenerate into corresponding solitary wave solutions, shock wave solutions and trigonometric function solutions.     

Bifurcation of travelling wave solutions for the modified dispersive water wave equation

The method of bifurcation of planar dynamical systems and method of numerical simulation of differential equations are employed to investigate the modified dispersive water wave equation. We obtain the parameter bifurcation sets that divide the parameter space into different regions which correspond to qualitatively different phase portraits. In different regions, different types of travelling solutions including solitary wave solutions, shock wave solutions and periodic wave solutions are simulated. Furthermore, with a generalized projective Riccati equation method, several new explicit exact solutions are obtained.     

The regularized Benjamin-Ono and BBM equations: Well-posedness and nonlinear stability

Nonlinear stability of nonlinear periodic solutions of the regularized Benjamin-Ono equation and the Benjamin-Bona-Mahony equation with respect to perturbations of the same wavelength is analytically studied. These perturbations are shown to be stable. We also develop a global well-posedness theory for the regularized Benjamin-Ono equation in the periodic and in the line setting. In particular, we show that the Cauchy problem (in both periodic and nonperiodic case) cannot be solved by an iteration scheme based on the Duhamel formula for negative Sobolev indices.     

Generalized 1D Jörgens theorem

We study the Cauchy problem for a 1D nonlinear wave equation on R. The nonlinearity can depend on the unknown function and its first order spatial derivative. Using the fixed point theorem we prove the existence of a classical solution. Moreover, the existence of periodic and almost periodic solutions are shown.     

Global singularity structure of weak solutions to 3-D semilinear dispersive wave equations with discontinuous initial data

We study the global singularity structure of solutions to 3-D semilinear wave equations with discontinuous initial data. More precisely, using Strichartz’ inequality we show that the solutions stay conormal after nonlinear interaction if the Cauchy data are conormal along a circle.     

TRAVELINGWAVE SOLUTIONS FOR A CLASS OF NONLINEAR DISPERSIVE EQUATIONS

The method of the phase plane is emploied to investigate the solitary and periodic traveling waves for a class of nonlinear dispersive partial differential equations. By using the bifurcation theory of dynamical systems to do qualitative analysis, all possible phase portraits in the parametric space for the traveling wave systems are obtained. It can be shown that the existence of a singular straight line in the traveling wave system is the reason why smooth solitary wave solutions converge to solitary cusp wave solution when parameters are varied. The different parameter conditions for the existence of solitary and periodic wave solutions of different kinds are rigorously determined.     

Periodic-like solutions of variable coefficient and Wick-type stochastic NLS equations

Variable coefficient and Wick-type stochastic nonlinear Schrödinger (NLS) equations are investigated. By using white noise analysis, Hermite transform and extended F-expansion method, we obtain a number of Wick versions of periodic-like wave solutions and periodic wave solutions expressed by various Jacobi elliptic functions for Wick-type stochastic and variable coefficient NLS equations, respectively. In the limit cases, the soliton-like wave solutions are showed as well. Since Wick versions of functions are usually difficult to evaluate, we get some nonWick versions of the solutions for Wick-type stochastic NLS equations in special cases.     

Stability of rarefaction wave for a macroscopic model derived from the Vlasov-Maxwell-Boltzmann system

In this article, we are concerned with the nonlinear stability of the rarefaction wave for a one-dimensional macroscopic model derived from the Vlasov-Maxwell-Boltzmann system. The result shows that the large-time behavior of the solutions coincides with the one for both the Navier-Stokes-Poisson system and the Navier-Stokes system. Both the time-decay property of the rarefaction wave profile and the influence of the electromagnetic field play a key role in the analysis.     

TRAVELING WAVE SOLUTIONS FOR A CLASS OF NONLINEAR DISPERSIVE EQUATIONS

The method of the phase plane is emploied to investigate the solitary and periodic travelingwaves for a class of nonlinear dispersive partial differential equations.By using the bifurcationtheory of dynamical systems to do qualitative analysis,all possible phase portraits in theparametric space for the traveling wave systems are obtained.It can be shown that the existenceof a singular straight line in the traveling wave system is the reason why smooth solitary wavesolutions converge to solitary cusp wave solution when parameters are varied.The differentparameter conditions for the existence of solitary and periodic wave solutions of different kindsare rigorously determined.     

Exact periodic wave solutions for the hKdV equation

In this paper, by using the Hirota bilinear method and the Jacobian theta functions for the higher order KdV equation, the existence of periodic wave solutions with one and two period are obtained. The asymptotic properties of the periodic wave solutions are analyzed in detail. It is shown that the well-known soliton solutions can be reduced from the periodic wave solutions.     

Construction of traveling wave solutions of the (2 + 1)-dimensional modified KdV-KP equation

Traveling wave solutions have played a vital role in demonstrating the wave character of nonlinear problems emerging in the field of mathematical sciences and engineering. To depict the nature of propagation of the nonlinear waves in nature, a range of nonlinear evolution equations has been proposed and investigated in the existing literature. In this article, solitary and traveling periodic wave solutions for the (2 + 1)-dimensional modified KdV-KP equation are derived by employing an ansatz method, named the enhanced (G′/G)-expansion method. For this continued equation, abundant solitary wave solutions and nonlinear periodic wave solutions, along with some free parameters, are obtained. We have derived the exact expressions for the solitary waves that arise in the continuum-modified KdV-KP model. We study the significance of parameters numerically that arise in the obtained solutions. These parameters play an important role in the physical structure and propagation directions of the wave that characterizes the wave pattern. We discuss the relation between velocity and parameters and illustrate them graphically. Our numerical analysis suggests that the taller solitons are narrower than shorter waves and can travel faster. In addition, graphical representations of some obtained solutions along with their contour plot and wave train profiles are presented. The speed, as well as the profile of these solitary waves, is highly sensitive to the free parameters. Our results establish that the continuum-modified KdV-KP system supports solitary waves having different shapes and speeds for different values of the parameters.     

Time periodic traveling wave solutions for periodic advection–reaction–diffusion systems

We study the existence, uniqueness, and asymptotic stability of time periodic traveling wave solutions to a class of periodic advection–reaction–diffusion systems. Under certain conditions, we prove that there exists a maximal wave speed c^?$c^{?}$ such that for each wave speed c≤c^?$c\le c^{?}$, there is a time periodic traveling wave connecting two periodic solutions of the corresponding kinetic system. It is shown that such a traveling wave is unique modulo translation and is monotone with respect to its co-moving frame coordinate. We also show that the traveling wave solutions with wave speed c≤c^?$c\le c^{?}$ are asymptotically stable in certain sense. In addition, we establish the nonexistence of time periodic traveling waves with speed c>c^?$c>c^{?}$.     

Existence and stability of modulating pulse solutions in Maxwell’s equations describing nonlinear optics

Modulating pulse solutions play a big rôle in modern long distance high speed communication. Such solutions consist of a traveling pulse-like envelope modulating an underlying electromagnetic wave. In this paper we show that under certain assumptions such solutions exist and are dynamically stable for the associated nonlinear partial differential equations, namely Maxwells integro-differential equations describing nonlinear optics. The analysis is worked out in detail for bulk media, and we discuss how the results extend to optical fibers and to parametrically forced systems.