Asymptotic stability of the fourth order $\phi^4$ kink for general perturbations in the energy space

TL;DR

This paper proves the orbital and asymptotic stability of the fourth order $^4$ kink solution in the energy space, extending stability results to a higher-order quantum field model with no spectral gap.

Contribution

It establishes the stability of the $^4$ kink for general perturbations in the energy space, a novel result for this higher-order dispersive model.

Findings

Proves orbital stability of the kink solution.

Establishes asymptotic stability under general perturbations.

Handles the lack of spectral gap in the linear operator.

Abstract

The Fourth order $\phi^4$ model generalizes the classical $\phi^4$ model of quantum field theory, sharing the same kink solution. It is also the dispersive counterpart of the well-known parabolic Cahn-Hilliard equation. Mathematically speaking, the kink is characterized by a fourth-order nonnegative linear operator with a simple kernel at the origin but no spectral gap. In this paper, we consider the kink of this theory, and prove orbital and asymptotic stability for any perturbation in the energy space.