# On asymptotic stability of moving kink for relativistic Ginzburg-Landau equation

## TL;DR

This paper proves the asymptotic stability of moving kink solutions in a one-dimensional relativistic Ginzburg-Landau equation, showing solutions decompose into a moving kink plus dispersive waves over time.

## Contribution

It establishes the asymptotic stability of moving kinks for the relativistic Ginzburg-Landau equation, extending understanding of soliton stability in relativistic wave models.

## Key findings

- Solutions near a kink converge to a moving kink plus dispersive waves.
- The remainder term decays in a global energy norm.
- Weighted energy decay results are crucial for the proof.

## Abstract

We prove the asymptotic stability of the moving kinks for the nonlinear relativistic wave equations in one space dimension with a Ginzburg-Landau potential: starting in a small neighborhood of the kink, the solution, asymptotically in time, is the sum of a uniformly moving kink and dispersive part described by the free Klein-Gordon equation. The remainder decays in a global energy norm. Crucial role in the proofs play our recent results on the weighted energy decay for the Klein-Gordon equations.