Periodic travelling wave solutions of a parabolic equation: a monotonicity result

This paper considers travelling wave solutions of a quasilinear parabolic equation corresponding to the propagation of a front in some striated medium. The striations, whose geometry play a crucial role on the type of solutions obtained, are supposed to be oblique and disposed in a periodic fashion. We show that the speed of such travelling waves depends in a monotonous way on the angle of inclination of the striations.     

一类退化-奇异抛物型方程的行波解Ⅱ

多孔介质动力学及生物群体动力学方程引起愈来愈多的人的注意,退化-奇异抛物型方程(u~m/m)_t=(u~k)_(xx)+u~nf(u)的行波解也成为人们关心的课题之一.Aronson对 m=k=1,u~nf(u)∈C~1[0,1]讨论了单调行波解的存在性与正则性,Hosono 对m=1,k≥2,n=0,f∈C~2[0,1],f(0)=f(1)=0,f′(0)<0,f″(0)(?)0,f′(1)<0且在(0,α)内 f(u)<0;在(α,1)内 f(u)>0,讨论了单调行波解的存在性与稳定     

Asymptotic behaviour of solutions of quasilinear evolutionary partial differential equations of parabolic type on unbounded spatial intervals

We study the long‐time behaviour of solutions to a quasilinear parabolic problem on a half‐line. The main result lies in showing the existence of a positive solution that converges to the travelling wave of solution to the stationary problem on the whole line. The main tools used here are the zero number theory and the concentration compactness principle. This result is a generalization of a result know for semilinear parabolic equations. Copyright © 2006 John Wiley & Sons, Ltd.     

On a certain class of a nonlinear evolution partial differential equations

An existence result and a priori bound for the solution of a second-order nonlinear parabolic equation are established. Also a generalized tanh-function method is used for constructing exact travelling wave solutions for the nonlinear diffusion equation of Fisher type originated from the considered partial differential equation. And new multiple soliton solutions are obtained.     

Strong and Weak Property of Travelling Wave Front Solutions for a Class of Degenerate Parabolic Equations

§ 1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　Ｉｎｔｈｉｓｐａｐｅｒｗｅｓｔｕｄｙｔｈｅｓｔｒｏｎｇａｎｄｗｅａｋｐｒｏｐｅｒｔｙｏｆｔｒａｖｅｌｌｉｎｇｗａｖｅｆｒｏｎｔｓｏｌｕｔｉｏｎｓ (ＴＷＦＳ)ｆｏｒｔｈｅｆｏｌｌｏｗｉｎｇｄｅｇｅｎｅｒａｔｅｐａｒａｂｏｌｉｃｅｑｕａｔｉｏｎ : ψ(ｕ) ｔ = 2 ｕ ｘ2 +ｆ(ｕ) . ( 1 .1 )　Ｗｅｌｏｏｋｆｏｒｓｏｌｕｔｉｏｎｓｏｆ( 1 .1 )ｏｆｔｈｅｆｏｒｍｕ(ｘ,ｔ) =ｑ(ｘ -ｃｔ) (ｃｉｓｔｈｅｗａｖｅｓｐｅｅｄ) ,ｗｈｅｒｅｑ( ξ) ,ξ=ｘ -ｃｔ,ａｓａｆｕｎｃｔｉｏｎｏｆａｓ…     

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.     

On the Dynamics of the Classical Transverse Field Ising Chain

We investigate stationary and travelling wave solutions of a special lattice differential equation in one space dimension. Depending on a parameter λ, results are given on the existence, shape and stability for these kind of solutions. The analysis of travelling wave solutions leads us to a functional differential equation with both forward and backward shifts. The existence of solutions of this equation will be proved by use of the implicit function theorem. In particular, we consider kink solutions and periodic solutions.     

Long-time behaviour of solutions of a class of nonlinear parabolic equations

In this paper, the long-time behaviour of solutions of a class of nonlinear parabolic equations is studied. It is shown that the solutions of initial-boundary value problem to the equations converge to a travelling wave solution of the equation or a self-similar solution of a Hamilton–Jacobi equation under certain conditions on initial and boundary values of the solutions.     

Stability of travelling waves in a model for conical flames in two space dimensions

This paper deals with the question of the stability of conical-shaped solutions of a class of reaction-diffusion equations in . One first proves the existence of travelling waves solutions with conical-shaped level sets, generalizing earlier results by Bonnet, Hamel and Monneau [SIAM J. Math. Anal. 31 (1999) 80-118; Comm. Partial Differential Equations 25 (2000) 769-819]. One then gives a characterization of the global attractor of these semilinear parabolic equations under some conical asymptotic conditions. Lastly, the global stability of the travelling waves solutions is proved.     

Long-time behaviour of solutions of a class of nonlinear parabolic equations

In this paper, the long-time behaviour of solutions of a class of nonlinear parabolic equations is studied. It is shown that the solutions of initial-boundary value problem to the equations converge to a travelling wave solution of the equation or a self-similar solution of a Hamilton–Jacobi equation under certain conditions on initial and boundary values of the solutions.     

Existence of Global and Periodic Solutions for Delay Equation with Quasilinear Perturbation

Delay parabolic problems have been studied by many authors. Some authors investigated more general delay problem (refer to [1], [2]), some investigated concrete delay partial differential equations. Recently, we have done some work on delay parabolic problem. We discussed semilinear parabolic delay problem and obtained some results on the existence of solutions. In particular the results on existence of periodic solutions are characteristic (see [3], [4], [5], [6]). The purpose of this paper is to study delay equation with quasilinear perturbation. We present the existence of global and periodic solutions of abstract evolution equations in Section 2. The abstract results are used to obtain the existence of global and periodic solutions of delay parabolic problem with quasilinear perturbation in Section 3. We make preparation for our investigation and give a generalization of Gronwall inequality (Lemma 1.3) which is used in next section.     

Dynamical understanding of loop soliton solution for several nonlinear wave equations

It has been found that some nonlinear wave equations have one-loop soliton solutions. What is the dynamical behavior of the so-called one-loop soliton solution? To answer this question, the travelling wave solutions for four nonlinear wave equations are discussed. Exact explicit parametric representations of some special travelling wave solutions are given. The results of this paper show that a loop solution consists of three different breaking travelling wave solutions. It is not one real loop soliton travelling wave solution.     

一类退化-奇异抛物型方程的行波解Ⅰ

多孔介质动力学方程吸引了众多人的注意,引起了数学家们的极大兴趣.退化-奇异抛物型方程的行波解也成为令人瞩目的问题.Aronson,Hosono,Atkinson,Engler,Grindron 和 Sleeman,Wang 和 Ye 等在这一领域中都有出色的工作.Aronson 对 m=k=1,u~nf(u)∈C~1[0,1],讨论了单调行波解的存在性.Hosono 对m=1,k≥2,n=0,f∈C~2[0,1],f′(0)<0,f″(0)(?)0,f′(1)<0,且存在α∈     

New travelling wave solutions to the Ostrovsky equation

In this work, new travelling wave solutions to the Ostrovsky equation are studied by employing the improved tanh function method. With this method, the Ostrovsky equation is reduced to the nonlinear ordinary differential equation and then the different types of exact solutions are derived based on the solutions of the Riccati equation. We will compare our solutions with those gained by the other methods.     

Kinks and travelling wave solutions for Burgers-like equations

In this work we develop a variety of Burgers-like equations. We show that these derived equations share some of the travelling wave solutions of the Burgers equation, and differ in some other solutions.     

The relationship among the solutions of two auxiliary ordinary differential equations

In a recent article [Phys. Lett. A 356 (2006) 124], Sirendaoreji extended their auxiliary equation method by introducing a new auxiliary ordinary differential equation (NAODE) and its 14 solutions. Then the author studied some nonlinear evolution equations (NLEEs) and got more exact travelling wave solutions. In this paper, we will show that the 14 solutions of the NAODE are actually the same as the solutions obtained by original auxiliary equation method, and they are only different in the form.     

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.     

Auxiliary equation method for solving nonlinear Wick-type partial differential equations

Using the solutions of an auxiliary differential equation, a direct algebraic method is described to construct several kinds of exact travelling wave solutions for some Wick-type nonlinear partial differential equations. By this method some physically important nonlinear equations are investigated and new exact travelling wave solutions are explicitly obtained. In addition, the links between Wick-type partial differential equations and variable coefficient partial differential equations are also clarified generally.     

New exact travelling wave solutions for some nonlinear evolution equations, Part II

By using new solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact travelling wave solutions for nonlinear evolution equation. By this method some nonlinear evolution equations are investigated and new exact travelling wave solutions are explicitly obtained with the aid of symbolic computation.     

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.