Recurrent solutions of the linearly coupled complex cubic-quintic Ginzburg-Landau equations

In this paper, we will establish the bounded solutions, periodic solutions, quasiperiodic solutions, almost periodic solutions, and almost automorphic solutions for linearly coupled complex cubic-quintic Ginzburg-Landau equations, under suitable conditions. The main difficulty is the nonlinear terms in the equations that are not Lipschitz-continuity, traditional methods cannot deal with the difficulty in our problem. We overcome this difficulty by the Galerkin approach, energy estimate method, and refined inequality technique.