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Volume 10, Number 1, 2020, Pages 297-313                                                                DOI:10.11948/20190152
Periodic and quasi-periodic solutions for the complex Swift-Hohenberg equation
Lufang Mi,Wenyan Cui,Honglian You
Keywords:Complex Swift-HohenbergC 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 solutions for the above equation.
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