Meromorphic Traveling Wave Solutions of the Complex Cubic-Quintic Ginzburg-Landau Equation

We look for singlevalued solutions of the squared modulus M of the traveling wave reduction of the complex cubic-quintic Ginzburg-Landau equation. Using Clunie??s lemma, we first prove that any meromorphic solution M is necessarily elliptic or degenerate elliptic. We then give the two canonical decompositions of the new elliptic solution recently obtained by the subequation method.     

广义的Zakharov方程和Ginzburg-Landau方程的精确解和行波解分支

获得了广义的Zakharov方程和Ginzburg-Landau方程的一些精确行波解,这些行波解有什么样的动力学行为,它们怎样依赖系统的参数？该文将利用动力系统方法回答这些问题,给出了两个方程的6个行波解的精确参数表达式．     

Solutions of the Ginzburg-Landau Equation of Interest in Shear Flow Transition

The Ginzburg-Landau equation may be used to describe the weakly nonlinear 2-dimensional evolution of a disturbance in plane Poiseuille flow at Reynolds number near critical. We consider a class of quasisteady solutions of this equation whose spatial variation may be periodic, quasiperiodic, or solitarywave- like. Of particular interest are solutions describing a transition from the laminar solution to finite amplitude states. The existence of these solutions suggests the existence of a similar class of solutions in the Navier-Stokes equations, describing pulses and fronts of instability in the flow.     

奇异摄动半线性抛物方程的数值解法

本文我们讨论了一维和二维奇异摄动半线性抛物方程,我们利用直线法和精确差分格式在非均匀网中上得到了数值解.而且还证明了关于ε的一阶一致收敛性.     

Proof of the Absence of Elliptic Solutions of the Cubic Complex Ginzburg-Landau Equation

We consider the cubic complex Ginzburg-Landau equation. Using Hone's method, based on formal Laurent-series solutions and the residue theorem, we prove the absence of elliptic standing-wave solutions of this equation. This result complements a result by Hone, who proved the nonexistence of elliptic traveling-wave solutions. We show that it is more efficient to apply Hone's method to a system of polynomial differential equations rather than to an equivalent differential equation. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 161–171, January, 2006.     

Approximation to Nonlinear Schrodinger Equation of the Complex Generalized Ginzburg-Landau Equation

In this paper, we prove that in the inviscid limit the solution of the generalized derivative Ginzburg-Landau equations converges to the solution of derivative nonlinear Schrödinger equation, we also give the convergence rates for the difference of the solution.     

The Perturbed Plane-Wave Solutions of the Cubic Schrödinger Equation

A detailed analysis is given to the solution of the cubic Schrödinger equation iq_t + q_xx + 2|q|²q = 0 under the boundary conditions as |x|→∞. The inverse-scattering technique is used, and the asymptotic state is a series of solitons. However, there is no soliton whose amplitude is stationary in time. Each soliton has a definite velocity and “pulsates” in time with a definite period. The interaction of two solitons is considered, and a possible extension to the perturbed periodic wave [q(x + T,t) = q(x,t) as |x|→∞] is discussed.     

On the Ginzburg-Landau Wave Equation

This note studies the global existence of strong solutions ofthe initial value problem of the Ginzburg-Landau wave equationon Rⁿ (n = 1, 2, 3). Current address: Department of Mathematics and Statistics, Universityof New Mexico, Albuquerque, NM 87131, USA     

Slowly Varying Solitary Wave Solutions of the Perturbed Korteweg-de Vries Equation Revisited

Using a multi-scale perturbation expansion we reconsider the slowly varying solitary wave asymptotic solution of the perturbed Korteweg-de Vries equation. The well-known results for the variation of the solitary-wave amplitude and the accompanying trailing tail are recovered. Here the analysis is carried through to second order so as to determine a general expression for the first-order speed correction. The result obtained here generalizes and improves previous results.     

2+1维Ginzburg-Landau方程的精确波解

分析研究了一个具有三次增益效应和五次耗散项的2+1维Ginzburg-Landau方程.利用同解变型法并结合一个高阶辅助方程的解,成功地取得了该方程的一些新的精确行波解.     

Local Dynamics of a Singularly Perturbed Second Order Equation with State-Dependent Delay

Mathematical Notes -     

复合Ginzburg-Landau方程的动力学行为分析

目前对非线性波动方程的研究大都仅限于静态波解,即所考虑的波解的波速、振幅、波宽都是不变的,考虑动态波解,以复合Ginzburg-Landau(CGLE)方程为研究对象,探讨其动力学行为.在假设示性函数的基础上,所研究的无穷维耗散系统转化为三维向量场,给出了简单分岔和Hopf分岔存在的条件,揭示了系统平衡点和极限环随系统参数的变化规律,分析了参数平面的不同区域中系统的相图特性,得到系统存在两种不同频率的周期解,此外还数值模拟了系统由倍周期分岔导致混沌的过程,揭示了系统的复杂性.     

Exponential Attractors for the Generalized Ginzburg-Landau Equation

Abstract In this paper, we establish the global fast dynamics for the derivative Ginzburg-Landau equation in two spatial dimensions. We show the squeezing property and the existence of finite dimensional exponential attractors for this equation * The author is supported by the Postdoctoral Foundation of China     

Pseudo Almost Periodic Solutions of a Singularly Perturbed Differential Equation with Piecewise Constant Argument

Under suitable assumptions, the existence and the uniqueness of the pseudo-almost periodic solution for a singularly perturbed differential equation with piecewise constant argument are obtained. In addition, the stability properties of these solutions are characterized by the construction of manifolds of initial data.     

Asymptotics of the Solutions to Singularly Perturbed Multidimensional Integral Equations

Asymptotics of solutions to the equations εh_ε + Rh_ε = f as ε → + 0 is studied for a class of multidimensional integral equations that are basic in estimation theory.     

Lin- Reissner-Tsien方程的自相似解

Lin-Reissner-Tsien方程描述了在跨音速近似下的不稳定跨音速流动.通过相似变换,将Lin-Reissner-Tsien方程简化为一个常微分方程.然后解析求解所得到的方程,并在某些情况下得到的正好是精确解.上述情况下无法得到精确解时,给出了数值模拟.还得到了行波解.     

The Time-Periodic Solution to a 2D Generalized Ginzburg-Landau Equation

In this paper, we study a 2D generalized Ginzburg-Landau equation with a periodic boundary condition. The existence and uniqueness of a time-periodic solution to this equation is proved.     

Constructing Solutions of an Ill-Posed Nonlinear Singularly Perturbed Problem for an Equation of Elliptic Type

Mathematical Notes -