The Thual-Fauve pulse: skew stabilization

TL;DR

This paper investigates the stabilization of localized pulse solutions in a perturbed quintic Ginzburg-Landau equation, providing analytical proofs and conditions for stabilization under small non-gradient perturbations.

Contribution

It offers a validated asymptotic analysis of pulse existence, eigenvalue expansions, and stabilization criteria for perturbed Ginzburg-Landau equations.

Findings

Existence of pulse solutions via validated asymptotics.

Eigenvalue and eigenvector expansions for the linearized operator.

A sufficient condition for pulse stabilization under perturbations.

Abstract

It is possible to choose the parameters of a real quintic Ginzburg-Landau equation so that it possesses localized pulse-like solutions; Thual and Fauve have observed numerically that these pulses are stabilized by perturbations destroying the gradient structure of the real equation. For parameters such that the real part of the equations possesses pulses with a large shelf, we prove the existence of pulses by validated asymptotics, we find the expansion of the small eigenvalues of the operator and of their corresponding eigenvectors, and we give a sufficient condition for stabilization. This condition is generalized to any small non-gradient quintic perturbation of Ginzburg-Landau.