PERIODIC AND QUASI-PERIODIC SOLUTIONS FOR THE COMPLEX SWIFT-HOHENBERG EQUATION

Abstract

In this paper, we consider the complex Swift-Hohenberg(CSH) equation $ \frac{\partial u}{\partial t} = \lambda u-(\alpha+\mathrm{i}\beta)\left(1+\frac{\partial^2}{\partial x^2}\right)^2u-(\sigma+\mathrm{i}\rho)|u|^2u $ subject to periodic boundary conditions. Using an infinite dimensional KAM theorem, we prove that there exist a continuous branch of periodic solutions and a Cantorian branch of quasi-periodic solutions for the above equation.