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

Abstract

We consider the complex Ginzburg-Landau equation with two pure-power nonlinearities and a damping term. After proving a general global existence result, we focus on the existence and stability of several periodic orbits, namely the trivial equilibrium, bound-states and solutions independent of the spatial variable. In particular, we construct bound-states either explicitly in the real line or through a bifurcation argument for a double eigenvalue of the Dirichlet-Laplace operator on bounded domains.

A note on bifurcations from eigenvalues of the Dirichlet-Laplacian with arbitrary multiplicity

“…However, for complex coefficients, this structure collapses and the construction of bound-states becomes far from being trivial. In the past decade, there have been some works in this direction [2,3,4,5], where one applies a bifurcation argument starting from the equation with λ, η P R. Another approach is to reduce the general (BS) (via a nontrivial transformation) to the real coefficients case. This approach, available for Ω " R ( [4]), seems innaplicable for any other domain.…”

“…Sketch of the proof. The proof is a simple generalization of the results in [5,7,8,11]. For part 1, one performs a Lyapunov-Schmidt reduction, decomposing φ as in (3).…”

“…Since }u} H 1 0 pΩq " Op q, a standard bootstrap argument yields }u} L 8 pΩq À Op q. Thus the linear operator Bptq referred in (5)…”