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 Periodic and Quasi-periodic Solutions for the Complex Swift-Hohenberg Equation Lufang Mi,Wenyan Cui,Honglian You Keywords:Swift-Hohenberg equation; periodic solution; quasi-periodic solution; normal form Abstract: In this paper, we consider the complex Swift-Hohenberg(CSH) equation $$\frac{\partial u}{\partial t}=\lambda u-(\alpha+\mathrm{i}\beta)\l(1+\frac{\partial^2}{\partial x^2}\r)^2u-(\sigma+\mathrm{i}\rho)|u|^2u$$ 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 solution for the above equation.