New explicit solutions for the Klein-Gordon equation with cubic nonlinearity

In this paper, we investigate Klein-Gordon equation with cubic nonlinearity. All explicit expressions of the bounded travelling wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain bell-shaped solitary wave solutions, kink-shaped solitary wave solutions and Jacobi elliptic function periodic solutions. Moreover, we point out the region which these periodic wave solutions lie in. We present the relation between the bounded travelling wave solution and the energy level h. We find that these periodic wave solutions tend to the corresponding solitary wave solutions as h increases or decreases. Finally, for some special selections of the energy level h, it is shown that the exact periodic solutions evolute into solitary wave solution.     

Qualitative Analysis and Travelling Wave Solutions for the Generalized Pochhammer–Chree Equation with a Dissipation Term

The purpose of this paper is to reveal the influence of dissipation on travelling wave solutions of the generalized Pochhammer–Chree equation with a dissipation term, and provides travelling wave solutions for this equation. Applying the theory of planar dynamical systems, we obtain ten global phase portraits of the dynamic system corresponding to this equation under various parameter conditions. Moreover, we present the relations between the properties of travelling wave solutions and the dissipation coefficient r of this equation. We find that a bounded travelling wave solution appears as a bell profile solitary wave solution or a periodic travelling wave solution when r= 0; a bounded travelling wave solution appears as a kink profile solitary wave solution when |r| > 0 is large; a bounded travelling wave solution appears as a damped oscillatory solution when |r| > 0 is small. Further, by using undetermined coefficient method, we get all possible bell profile solitary wave solutions and approximate damped oscillatory solutions for this equation. Error estimates indicate that the approximate solutions are meaningful.     

Modulational Stability of Travelling Waves in 2D Anisotropic Systems

The modulational stability of travelling waves in 2D anisotropic systems is investigated. We consider normal travelling waves, which are described by solutions of a globally coupled Ginzburg–Landau system for two envelopes of left- and right-travelling waves, and oblique travelling waves, which are described by solutions of a globally coupled Ginzburg–Landau system for four envelopes associated with two counterpropagating pairs of travelling waves in two oblique directions. The Eckhaus stability boundary for these waves in the plane of wave numbers is computed from the linearized Ginzburg–Landau systems. We identify longitudinal long and finite wavelength instabilities as well as transverse long wavelength instabilities. The results of the stability calculations are confirmed through numerical simulations. In these simulations we observe a rich variety of behaviors, including defect chaos, elongated localized structures superimposed to travelling waves, and moving grain boundaries separating travelling waves in different oblique directions. The stability classification is applied to a reaction–diffusion system and to the weak electrolyte model for electroconvection in nematic liquid crystals.      

Propagation of internal waves in a three-layer liquid

The propagation o f internal harmonic waves in a three-layer liquid is examined in a linear approximation for arbitrary wave numbers. A travelling wave sohaion is obtained and a dispersion equation is derived. The roots of the dispersion equation and the dependence of the solution on the ratio of the densities of the liquids are examined in detail. The phase and group velocities are plotted as functions of wavelength and graphs of the various wave modes are presented. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 170–174, 1999.     

New explicit solutions for (2 + 1)-dimensional soliton equation

In this Letter, we study (2 + 1)-dimensional soliton equation by using the bifurcation theory of planar dynamical systems. Following a dynamical system approach, in different parameter regions, we depict phase portraits of a travelling wave system. Bell profile solitary wave solutions, kink profile solitary wave solutions and periodic travelling wave solutions are given. Further, we present the relations between the bounded travelling wave solutions and the energy level h. Through discussing the energy level h, we obtain all explicit formulas of solitary wave solutions and periodic wave solutions.     

Periodic travelling waves in an artificial neural network

In a study (Szekely, Acta Physiol. Hung. 27 (1965), pp. 285–289) of the locomotion of salamanders, it is observed that a ‘doubly periodic travelling wave solution’ of a logical neural network can be used to explain a dynamic pattern of movements. We show here that a relatively simple (nonlogical) artificial neural network can also be built and necessary and sufficient conditions for the existence of doubly periodic travelling wave solutions can be found. It is hoped that our investigation will set some foundation in the future design of other artificial neural networks that also allow periodic travelling wave solutions.     

Asymptotic stability of viscous shock wave profiles for viscous conservation laws

This paper is concerned with the asymptotic stability of travelling wave solution to the two-dimensional steady isentropic irrotational flow with artificial viscosity. We prove that there exists a unique travelling wave solution up to a shift to the system if the end states satisfy both the Rankine–Hugoniot condition and Lax's shock condition, and that the travelling wave solution is stable if the initial disturbance is small.     

The Bäcklund transformations and abundant explicit exact solutions for the AKNS–SWW equation

The Bäcklund transformations and abundant explicit exact solutions to the AKNS shallow water wave equation are obtained by combining the extended homogeneous balance method with the extended hyperbolic function method. The solutions obtained admit of multiple arbitrary parameters. These solutions include (a) a compound of the rational fractional function and a linear function, (b) a compound of solitary wave solution and a linear function, (c) a compound of the singular travelling wave solutions and a linear function, and (d) a compound of the periodic wave solutions of triangle function and a linear function. In special cases, we can obtain a series of soliton solutions, singular travelling wave solutions, periodic travelling wave solutions, and rational fractional function solution. In addition to re-deriving some known solutions in a systematic way, some brand-new exact solutions are also established.     

Periodic Travelling Waves and Compactons in Granular Chains

We study the propagation of an unusual type of periodic travelling waves in chains of identical beads interacting via Hertz’s contact forces. Each bead periodically undergoes a compression phase followed by free flight, due to special properties of Hertzian interactions (fully nonlinear under compression and vanishing in the absence of contact). We prove the existence of such waves close to binary oscillations, and numerically continue these solutions when their wavelength is increased. In the long wave limit, we observe their convergence towards shock profiles consisting of small compression regions close to solitary waves, alternating with large domains of free flight where bead velocities are small. We give formal arguments to justify this asymptotic behavior, using a matching technique and previous results concerning solitary wave solutions. The numerical finding of such waves implies the existence of compactons, i.e. compactly supported compression waves propagating at a constant velocity, depending on the amplitude and width of the wave. The beads are stationary and separated by equal gaps outside the wave, and each bead reached by the wave is shifted by a finite distance during a finite time interval. Below a critical wave number, we observe fast instabilities of the periodic travelling waves, leading to a disordered regime.     

A New Evans Function for Quasi-Periodic Solutions of the Linearised Sine-Gordon Equation

We construct a new Evans function for quasi-periodic solutions to the linearisation of the sine-Gordon equation about a periodic travelling wave. This Evans function is written in terms of fundamental solutions to a Hill’s equation. Applying the Evans-Krein function theory of Kollár and Miller (SIAM Rev 56(1):73–123, 2014) to our Evans function, we provide a new method for computing the Krein signatures of simple characteristic values of the linearised sine-Gordon equation. By varying the Floquet exponent parametrising the quasi-periodic solutions, we compute the linearised spectra of periodic travelling wave solutions of the sine-Gordon equation and track dynamical Hamiltonian–Hopf bifurcations via the Krein signature. Finally, we show that our new Evans function can be readily applied to the general case of the nonlinear Klein–Gordon equation with a non-periodic potential.

     

广义特殊Tzitzeica-Dodd-Bullough类型方程的行波解

本文研究了广义特殊Tzitzeica-Dodd-Bullough类型方程,利用动力系统分支理论方法,证明该方程存在周期行波解,无界行波解和破切波解,并求出了一些用参数表示的显示精确行波解.     

Travelling waves and dispersion relation in the spatial unfolding of a periodic orbitOndes progressives et relation de dispersion dans le déploiement spatial d'une orbite périodique

For a partial differential equation in spatial dimension one, admitting a spatially homogeneous time periodic solution, we show the generic existence, close to this solution, of a one-parameter family of travelling waves parametrized by their wave number k (k=0 corresponding to the spatially homogeneous initial solution). The argument is elementary and relies on a direct application of singular perturbation theory (Fenichel's global center manifold theorem). To cite this article: E. Risler, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 833–838.     

Explicit periodic travelling waves for a discrete lambda–omega reaction–diffusion system

In 1973, Kopell and Howard introduced a λ–ω reaction–diffusion system and found an explicit family of periodic travelling wave solutions lying on circles with radius less than 1. Since λ–ω systems represent universal models for studying chemical processes, and onset of turbulent behaviour, etc., explicit solutions of λ–ω systems with delays or discrete λ–ω systems can be of further help when the only method for obtaining other solutions is through numerical computation. There are now much investigations of various λ–ω systems. However, it is of interest to note that none attempts to find explicit travelling wave solutions. In this paper, we investigate the existence of explicit solutions for the simplest Euler scheme of a λ–ω system with delays or advancements which is described as a coupled pair of partial difference equations. We are able to provide necessary as well as sufficient conditions for the existence of numerical periodic travelling wave solutions. Additionally, we also provide some examples to show that our explicit solutions are qualitatively different from those found by Kopell and Howard and hence they may be of interests for specialists in the area of reaction–diffusion systems.     

Asymptotic pattern for a periodic lattice model with age structure and global interaction

In this paper, we derive a time-periodic lattice model for a single species in a patchy environment, which has age structure and an infinite number of patches connected locally by diffusion. By appealing to the theory of asymptotic speed of propagation and monotonic periodic semiflows for travelling waves, we establish the existence of periodic travelling wave and spreading speed of the model.     

Effect of variable viscosity on the peristaltic transport of a Newtonian fluid in an asymmetric channel

The effect of variable viscosity on the peristaltic flow of a Newtonian fluid in an asymmetric channel has been discussed. Asymmetry in the flow is induced due to travelling waves of different phase and amplitude which propagate along the channel walls. A long wavelength approximation is used in the flow analysis. Closed form analytic solutions for velocity components and longitudinal pressure gradient are obtained. The study also shows that, in addition to the effect of mean flow parameter, the wave amplitude also effect the peristaltic flow. This effect is noticeable in the pressure rise and frictional forces per wavelength through numerical integration.     

Numerical computation of sharp travelling waves of a degenerate diffusion–reaction equation arising in biofilm modelling

Many degenerate diffusion–reaction equations permit sharp travelling wave solutions that describe the propagation of an interface with finite speed. If the equation is at least double degenerate, the derivative of the travelling wave solution can blow up at the interface, which poses considerable challenges for the computation of the travelling wave speed. We propose a numerical method for this problem that is based on the idea to approximate the multiple degenerate problem by a family of simple degenerate problems. For the latter we propose an interval-bracketing algorithm based on the theory of Sanchez-Garduno and Maini. The travelling wave speed of the original problem is obtained as the limit of the travelling wave speeds of the auxiliary problems. The performance of the method is investigated in a numerical simulation experiment for a problem that arises in the mathematical modelling of biofilm processes.     

一类一维阵列的孤波特征

该文研究了一类由二自由度可积哈密顿系统构成的一维阵列的行波解,发现在长波极限下,问题可约化为分析哈密顿系统在扰动下的同异宿轨道的情形．当无扰系统具有共振时,利用能量──相方法,得到该系统存在同、异宿到不动点和周期轨的充分条件,在该条件下相应地一维阵列存在一组具有孤波特征的行波,同时给出了一个Ｎ脉冲孤立子波的例子．     

A multi-dimensional bistable nonlinear diffusion equation in a periodic medium

In this work we study the existence of wave solutions for a scalar reaction-diffusion equation of bistable type posed in a multi-dimensional periodic medium. Roughly speaking our result states that bistability ensures the existence of waves for both balanced and unbalanced reaction term. Here the term wave is used to describe either pulsating travelling wave or standing transition solution. As a special case we study a two-dimensional heterogeneous Allen–Cahn equation in both cases of slowly varying medium and rapidly oscillating medium. We prove that bistability occurs in these two situations and we conclude to the existence of waves connecting \(u = 0\) and \(u = 1\). Moreover in a rapidly oscillating medium we derive a sufficient condition that guarantees the existence of pulsating travelling waves with positive speed in each direction.     

Phase differences in reaction-diffusion-advection systems and applications to morphogenesis

The authors study the effect of advection on reaction-diffusionpatterns. It is shown that the addition of advection to a two-variablereaction–diffusion system with periodic boundary conditionsresults in the appearance of a phase difference between thepatterns of the two variables which depends on the differencebetween the advection coefficients. The spatial patterns movelike a travelling wave with a fixed velocity which depends onthe sum of the advection coefficients. By a suitable choiceof advection coefficients, the solution can be made stationaryin time. In the presence of advection a continuous change inthe diffusion coefficients can result in two Turing-type bifurcationsas the diffusion ratio is varied, and such a bifurcation canoccur even when the inhibitor species does not diffuse. It isalso shown that the initial mode of bifurcation for a givendomain size depends on both the advection and diffusion coefficients.These phenomena are demonstrated in the numerical solution ofa particular reaction–diffusion system, and finally apossible application of the results to pattern formation inDrosophila larvae is discussed.     

Bifurcation of travelling wave solutions for （2＋1）-dimension nonlinear dispersive long wave equation

In this paper,the bifurcation of solitary,kink,anti-kink,and periodic waves for （2＋1）-dimension nonlinear dispersive long wave equation is studied by using the bifurcation theory of planar dynamical systems.Bifurcation parameter sets are shown,and under various parameter conditions,all exact explicit formulas of solitary travelling wave solutions and kink travelling wave solutions and periodic travelling wave solutions are listed.