Ginzburg-Landau方程的周期解

The time-periodic solution for a Ginzburg-Landau Equation

利用伽辽金方法、Leray-Schauder不动点原理和先验估计,证明了在带周期外力扰动和周期边界条件的影响下,非线性发展Ginzburg-Landau方程ut=(l+iα)Δu-(k+iβ)u2u+γ+f的时间周期解,其中f(t,x)是一个关于时间变量t的以ω为周期的函数.此方程具有非线性项u2u,给周期解的研究带来技术性困难,文章比较系统地解决了这个问题,给Ginzburg-Landau方程的研究作了补充和完善,同时也给与该方程联系的有限振幅不稳定波比如一些流体动力学系统,Rayleigh-B啨nard对流等方面的研究带来某些理论上的解释.

We study nonlinear evolution Ginzburg-Landau Equation ut = （λ ＋ iα）△u - （k ＋ iβ） ｜u｜^2u ＋ γ ＋f with periodic boundary condition and periodic outside force ,where f（t ,x） is an ω- periodic function of time variable t. The existence of a time-periodic solution is proved by the Galerkin method, Leray-Schauder fixed point theorem and priori estimates. The equation owns the nonlinear ｜u｜^2 u bring quite large technique difficulty for periodic solution investigation, the question was systematically solved in this paper, which complemented and consummated the study of Ginzburg-Landau Equation, at the same time,which of instability waves involved equation ,such as fluids brought some theory explanation for the study of finite amplitude dynamics system, Rayleigh-Benard convection,and so on .