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…     

Almost periodic solutions of discrete Volterra equations

For discrete Volterra equations with or without delay, we obtain several results concerning almost periodic solutions and asymptotically almost periodic solutions under certain conditions. We also investigate the relations among solutions of equations discussed and give an example to illustrate our results.     

Orbital stability of periodic traveling wave solutions to the generalized Long-Short wave equations

This paper investigates the orbital stability of periodic traveling wave solutions to the generalized Long-Short wave equations $\left\{\begin{array}{l}i\varepsilon_{t}+\varepsilon_{xx}=n\varepsilon+\alpha|\varepsilon|^{2}\varepsilon,\\n_{t}=(|\varepsilon|^{2})_{x}, x\in R.\end{array} \right.$ Firstly, we show that there exist a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period $L$ for the generalized Long-Short wave equations. Then, combining the classical method proposed by Benjamin, Bona et al., and detailed spectral analysis given by using Lame equation and Floquet theory, we show that the dnoidal type periodic wave solution is orbitally stable by perturbations with period $L$. As the modulus of the Jacobian elliptic function $k\rightarrow 1$, we obtain the orbital stability results of solitary wave solution with zero asymptotic value for the generalized Long-Short equations. In particular, as $\alpha=0$, we can also obtain the orbital stability results of periodic wave solutions and solitary wave solutions for the long-short wave resonance equations. The results in the present paper improve and extend the previous stability results of long-shore wave equations and its extension equations.     

Existence of asymptotically almost periodic solutions for damped wave equations

In this paper, a class of nonlinear damped wave equations of the form αu^?(t)+u^″(t)=βAu(t)+γAu^′(t)+f(t,u(t)), t?0, satisfying αβ<γ with prescribed initial conditions are studied. Some sufficient conditions are established for the existence and uniqueness of an asymptotically almost periodic solution. These results have significance in the study of vibrations of flexible structures possessing internal material damping. Finally, an example is presented to illustrate the feasibility and effectiveness of the results.     

Gabor systems on discrete periodic sets

Due to its good potential for digital signal processing, discrete Gabor analysis has interested some mathematicians. This paper addresses Gabor systems on discrete periodic sets, which can model signals to appear periodically but intermittently. Complete Gabor systems and Gabor frames on discrete periodic sets are characterized; a sufficient and necessary condition on what periodic sets admit complete Gabor systems is obtained; this condition is also proved to be sufficient and necessary for the existence of sets E such that the Gabor systems generated by χ _E are tight frames on these periodic sets; our proof is constructive, and all tight frames of the above form with a special frame bound can be obtained by our method; periodic sets admitting Gabor Riesz bases are characterized; some examples are also provided to illustrate the general theory. This work was supported by National Natural Science Foundation of China (Grant No. 10671008), Beijing Natural Science Foundation (Grant No. 1092001), PHR (IHLB) and the project sponsored by SRF for ROCS, SEM of China     

New exact periodic wave solutions for the Dullin-Gottwald-Holm equation

In this paper, the Dullin-Gottwald-Holm equation is studied using semi-inverse method and integral bifurcation method. New periodic waves such as peakon-like periodic wave, compacton-like periodic wave and singular periodic wave are found and their dynamical behaviors and certain strange phenomena are explained using the proposed criterion. The exact parametric representations of these waves are also presented.     

某些发展方程的渐近波速和行波解研究简介

非线性发展方程渐近波速和行波解的存在性是发展方程理论研究中两个重要课题,因其具有强烈的实际背景和对数学理论提出的许多挑战性问题,正引起愈来愈多数学家的广泛关注,近30多年来,特别是近10年,对一些典型类型发展方程行波解及其相关的最小波速、渐近波速的理论研究得到了迅速发展,涌现出很多代表性的成果,本文力求总结这一领域的最新进展,向读者展示相关问题发展的背景、线索、脉络和重要的研究方法,以期待研究的进一步深入。     

Positive periodic solutions for a class of delay differential equations

In this paper, we employ fixed point theorem and functional equation theory to study the existence of positive periodic solutions of the delay differential equation

x^′(t)=α(t)x(t)-β(t)x²(t)+γ(t)x(t-τ(t))x(t).     

Propagation failure in discrete periodic media

We study the periodic lattice dynamical systems with bistable nonlinearity. We use Moser's theorem to show that there exist infinitely many stationary solutions when one of the migration coefficients is sufficiently small. Moreover, we prove that the propagation failure occurs when both migration coefficients are sufficiently small.     

Differential equations with almost periodic or conditionally periodic coefficients: Recurrence and reducibility

Recurrence of an individual trajectory of a linear nonautonomous differential equation on a compact Lie group for the case of an almost periodic or conditionally periodic dependence of the right-hand side on time is proved. The relation between recurrence and reducibility is examined.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 229–237, August, 1998.This research was supported by the Russian Foundation for Basic Research under grants No. 96-01-00378 and No. 96-15-96072.     

Wavelet decomposition of the space of discrete periodic splines

An orthogonal basis for the spaceS _r ^m of discrete periodic splines is constructed. The wavelet decomposition of the spaceS _r ^m form=2 ^t is obtained using this basis. We derive recurrence formulas for the transformation from the decomposition with respect to the orthogonal basis to the wavelet decomposition, as well as recurrence formulas for the inverse transformation. Translated fromMatematicheskie Zametki, Vol. 67, No. 5, pp. 712–720, May, 2000.     

Cantor families of periodic solutions for completely resonant wave equations

We present recent existence results of Cantor families of small amplitude periodic solutions for completely resonant nonlinear wave equations. The proofs rely on the Nash-Moser implicit function theory and variational methods.      

Diophantine approximation, Bessel functions and radially symmetric periodic solutions of semilinear wave equations in a ball

The aim of this paper is to consider the radially-symmetric periodic-Dirichlet problem on for the equation

where is the classical Laplacian operator, and denotes the open ball of center and radius in When is a sufficiently large irrational with bounded partial quotients, we combine some number theory techniques with the asymptotic properties of the Bessel functions to show that is not an accumulation point of the spectrum of the linear part. This result is used to obtain existence conditions for the nonlinear problem.

     

Homoclinic solutions in periodic difference equations with saturable nonlinearity

In this paper, a periodic difference equation with saturable nonlinearity is considered. Using the linking theorem in combination with periodic approximations, we establish sufficient conditions on the nonexistence and on the existence of homoclinic solutions. Our results not only solve an open problem proposed by Pankov, but also greatly improve some existing ones even for some special cases.     

Infinite propagation speed for the Degasperis-Procesi equation

We prove that any nontrivial classical solution of the Degasperis-Procesi equation will not have compact support if its initial data has this property.     

一类Lotka-Volterra竞争模型的最小波速

研究了一类带有单稳非线性项的三物种竞争系统行波解最小波速的速度选择.首先利用比较方法,通过构造适当的上下解,建立了最小波速的速度选择机制.然后证明了物种的竞争系数关于速度选择的阈值结果,并得到了阈值的估计.最后借助数值模拟说明所得结果推广了已有文献的相关工作.     

Explicit asymptotic limits for a class of discrete Volterra equations

This paper presents explicit asymptotic limits for some solutions of a special kind of Volterra difference equations and exhibits a class of equations having asymptotic equilibrium.     

ON THE DEPENDENT PROPERTY OF SOLUTIONS FOR HIGHER ORDER PERIODIC DIFFERENTIAL EQUATIONS

This article investigates the property of linearly dependence of solutions f(z) and f(z 2πi)for higher order linear differential equations with entire periodic coefficients.