Quasi-periodic Solutions For The Complex Ginzburg-Landau Equation With Quas-periodic Forcing

The classical KAM theory which is constructed by three famous math-ematicians Kolmogorov, Arnold, Moser in the last1950s, is one of the most important mathematics achievements in the twentieth century. And studying Hamilton PDE with KAM theory is started in the late1980s. Ginzburg-Landau equation, named after Ginzburg and Landau in1950, is a mathematical equa-tion used to describe superconductivity. It plays a great role in many fields such as Dynamics, Quantum mechanics and Thermodynamics. We investigate the quasi-periodic solutions for the complex Ginzburg-Landau equation with quasi-periodic forcing (the frequency ω=(ω1,ω2,...,ωn)) in its linear ter-m with KAM theory. By making reduction to the original equation and then constructing a KAM iterative sequence to the new system, we prove that there are m+2-dimensional invariant torus for the reduced equation. So we get the quasi-periodic solutions for the original equation.This paper is divided into three chapters. In the first one, we briefly introduce the KAM theory and make a description of Ginzburg-Landau e-quation and give a brief summary of important developments of this prob-lem. In the following we make reduction to the linear part of the complex Ginzburg-Landau equation. Then performing a normal form and action an-gel variable to satisfy the condition of the KAM iteration. In the last chapter, we prove the main result with a constructed KAM iteration. It is proven that the existence of m+2-dimensional invariant torus for the original equation. As an important part of the KAM iteration we make measure estimate the last part of the paper.