Abstract

We consider a periodic boundary value problem for a nonlocal Ginzburg–Landau equation in its weakly dissipative version. The existence, stability, and local bifurcations of one-mode periodic solutions are studied. It is shown that in a neighborhood of one-mode periodic solutions there may exist a three-dimensional local attractor filled with spatially inhomogeneous time-periodic solutions. Asymptotic formulas for these solutions are obtained. The results are based on using and developing methods of the theory of infinite-dimensional dynamical systems. In a special version of the partial integro-differential equation considered, we study the existence of a global attractor. Solution in the form of series are obtained for this version of the nonlinear boundary value problem.

中文翻译：

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非局部Ginzburg-Landau方程的弱耗散版本的不变流形

摘要

我们考虑一个弱耗散形式的非局部Ginzburg-Landau方程的周期边值问题。研究了单模周期解的存在性，稳定性和局部分歧。结果表明，在单模周期解的附近，可能存在一个三维局部吸引子，充满了空间上不均匀的时间周期解。获得了这些解决方案的渐近公式。结果是基于无限维动力系统理论的使用和发展方法。在所考虑的偏积分-微分方程的一个特殊版本中，我们研究了整体吸引子的存在。对于该版本的非线性边值问题，获得了级数形式的解。