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In this paper, we consider the complex Swift-Hohenberg(CSH) equation ∂u ∂t = λu − (α + iβ) ( 1 + ∂ 2 ∂x2 )2 u − (σ + iρ)|u|u 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.

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