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

Abstract

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.

Polynomial Complex Ginzburg-Landau equations in almost periodic spaces

“…Almost periodic spaces were introduced by Bohr [9] and further developed by Stepanoff [20], Weyl [21] and Besicovitch [4], [3]. These spaces are well-studied for the Complex Ginzburg-Landau equations with different nonlinearities [18], [15]. We consider u 0 with specific irrational phases, and we prove that the time evolution by (1) maintains the same phases.…”

Periodic and almost periodic solutions for the damped Korteweg‐de Vries equation

“…In many partial differential equations, the study of periodic and almost periodic solutions has attracted considerable interest: Periodic solution: Navier‐Stokes equation, wave equation, Swift‐Hohenberg equation, Hamilton‐Jacobi equation, etc. Almost periodic solution: parabolic equation, wave equation, Ginzburg‐Landau equation, viscous Camassa‐Holm equation, Schrödinger equation, thermoelastic plate equation, weakly dissipated hybrid system, ect. …”

Recurrent solutions of the Schrödinger-KdV system with boundary forces

“…Proof of Theorem 1.3. To prove Theorem 1.3, we need to borrow some ideas developed in [8,9,10,13]. Case I.…”