Existence of periodic traveling wave solutions for the Ostrovsky equation

We are concerned with the Ostrovsky equation, which is derived from the theory of weakly nonlinear long surface and internal waves in shallow water under the presence of rotation. On the basis of the variational method, we show the existence of periodic traveling wave solutions. Copyright © 2008 John Wiley & Sons, Ltd.     

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…     

Ground state solutions of the periodic discrete coupled nonlinear Schrödinger equations

In this paper, by using the Nehari manifold approach in combination with periodic approximations, we obtain the sufficient conditions on the existence of the nontrivial ground state solutions of the periodic discrete coupled nonlinear Schrödinger equations. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Existence And Uniqueness Of A Traveling Wave Front Of A Model Equation In Synaptically Coupled Neuronal Networks

Consider the model equation in synaptically coupled neuronal networks@u@t+ m(u − n)= ( − au) Z 10(c) ZRK(x − y)H uy, t −1c|x − y| − dydc+ ( − bu) Z 10( ) ZRW(x − y)H(u(y, t − ) − )dyd.In this model equation, u = u(x, t) stands for the membrane potential of a neuron at position x andtime t. The kernel functions K 0 and W 0 represent synaptic couplings between neurons insynaptically coupled neuronal networks. The Heaviside step function H = H(u − ) represents thegain function and it is defined by H(u − ) = 0 for all u &lt; , H(0) = 12 and H(u − ) = 1 for allu &gt; . The functions and represent probability density functions. The function f(u) m(u − n)represents the sodium current, where m &gt; 0 is a positive constant and n is a real constant. Theconstants a 0, b 0, 0, 0 and &gt; 0 represent biological mechanisms. This model equationis motivated by previous models in synaptically coupled neuronal networks.We will couple together intermediate value theorem, mean value theorem and many techniquesin dynamical systems to prove the existence and uniqueness of a traveling wave front of this modelequation. One of the most interesting and difficult parts is the proof of the existence and uniquenessof the wave speed. We will introduce several auxiliary functions and speed index functions to provethe existence and uniqueness of the front and the wave speed.     

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).     

Existence and asymptotic behaviour of solutions to first- and second-order difference equations with periodic forcing

By using previous results of Djafari Rouhani for non-expansive sequences in Refs (Djafari Rouhani, Ergodic theorems for nonexpansive sequences in Hilbert spaces and related problems, Ph.D. Thesis, Yale University, Part I (1981), pp. 1–76; Djafari Rouhani, J. Math. Anal. Appl. 147 (1990), pp. 465–476; Djafari Rouhani, J. Math. Anal. Appl. 151 (1990), pp. 226–235), we study the existence and asymptotic behaviour of solutions to first-order as well as second-order difference equations of monotone type with periodic forcing. In the first-order case, our result extends to general maximal monotone operators, the discrete analogue of a result of Baillon and Haraux (Rat. Mech. Anal. 67 (1977), 101–109) proved for subdifferential operators. In the second-order case, our results extend among other things, previous results of Apreutesei (J. Math. Anal. Appl. 288 (2003), 833–851) to the non-homogeneous case, and show the asymptotic convergence of every bounded solution to a periodic solution.     

Traveling wave solutions for fractional partial differential equations arising in mathematical physics by an improved fractional Jacobi elliptic equation method

In this paper, combining with a new generalized ansätz and the fractional Jacobi elliptic equation, an improved fractional Jacobi elliptic equation method is proposed for seeking exact solutions of space‐time fractional partial differential equations. The fractional derivative used here is the modified Riemann‐Liouville derivative. For illustrating the validity of this method, we apply it to solve the space‐time fractional Fokas equation and the the space‐time fractional BBM equation. As a result, some new general exact solutions expressed in various forms including the solitary wave solutions, the periodic wave solutions, and Jacobi elliptic functions solutions for the two equations are found with the aid of mathematical software Maple. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Existence of periodic solutions for a class of second-order discrete Hamiltonian systems

By using the variant fountain theorem, we study the existence of periodic solutions for a class of superquadratic non-autonomous second-order discrete Hamiltonian systems.     

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.