Some stability results for the complex Ginzburg–Landau equation

Abstract

Using some classical methods of dynamical systems, stability results and asymptotic decay of strong solutions for the complex Ginzburg-Landau equation (CGL),with a > 0, α, b, β, k ∈ R, are obtained. Moreover, we show the existence of bound-states under certain conditions on the parameters and on the domain. We conclude with the proof of asymptotic stability of these bound-states when Ω = R and −k large enough.

A generalized complex Ginzburg-Landau equation: Global existence and stability results

“…In [6] it is proven that, for Ω bounded, the equation (B-S) (with k " 0) has a solution pω, uq P RˆH 1 0 pΩq bifurcating from u " 0 if σ is sufficiently small and cos θ cos γ ą 0. A similar result is obtained in [8] where the aim was to trade the freedom in k for the freedom in σ. The reference [9] focuses on a bifurcation argument starting from the ground-state solution of the nonlinear Schrödinger equation, for both Ω bounded and the whole space (under some radial assumptions).…”

“…In this paper, we extend some results of global existence of solutions and their stability and also the existence of standing wave solutions in one dimension, previously exposed for the complex Ginzburg-Landau equation in [8], where only one nonlinear term was present. Moreover, we prove the existence of standing waves in bounded domains through a bifurcation argument applied to double eigenvalues of the Dirichlet-Laplace operator, which is new even in the context of (CGL).…”

“…The existence of standing waves for the complex Ginzbourg-Landau equation remains a largely open problem. Before we proceed, we rewrite the generalized complex Ginzburg-Landau equation in its trigonometric form, following the notations of [6,8]: u t " e iθ ∆u`e iγ1 |u| σ1 u`χe iγ2 |u| σ2 u`ku, (gCGL*) where´π{2 ă θ ă π{2,´π ă γ 1 , γ 2 ď π, k P R, χ "˘1. Then one may look for solutions of (gCGL*) in the form u " e iωt φpxq, where φ P H 1 pΩq is a solution of the elliptic equation…”

“…For k " 0, the existence of standing wave solutions is already known in some particular cases : θ "˘γ "˘π{2, which corresponds to the nonlinear Schrödinger equation or ω " 0 (stationary solutions). Outside of these cases, we refer [6,8,9], where the implicit function theorem is used to obtain the existence of standing waves of (B-S) for χ " 0 and several constraints on the remaining parameters. In [6] it is proven that, for Ω bounded, the equation (B-S) (with k " 0) has a solution pω, uq P RˆH 1 0 pΩq bifurcating from u " 0 if σ is sufficiently small and cos θ cos γ ą 0.…”

“…Remark 2. In [8], the uniqueness and stability of bound-states defined on R was studied. The same arguments may be applied in our framework without any extra difficulty.…”