{\phi}^4 Solitary Waves in a Parabolic Potential: Existence, Stability, and Collisional Dynamics

TL;DR

This paper investigates how an external parabolic potential affects the existence, stability, and collision dynamics of kink solutions in a {} model, revealing that all stationary solutions are unstable and that trap parameters influence collision outcomes.

Contribution

It introduces a {} model with a parabolic potential, analyzing the stability and dynamics of kink solutions, and highlights the impact of trap strength on collision phenomena.

Findings

All stationary kink solutions are unstable.

Kink-antikink collision characteristics depend on trap parameters.

The trap influences critical velocity and multi-bounce windows.

Abstract

We explore a {\phi}^4 model with an added external parabolic potential term. This term dramatically alters the spectral properties of the system. We identify single and multiple kink solutions and examine their stability features; importantly, all of the stationary structures turn out to be unstable. We complement these with a dynamical study of the evolution of a single kink in the trap, as well as of the scattering of kink and anti-kink solutions of the model. We see that some of the key characteristics of kink-antikink collisions, such as the critical velocity and the multi-bounce windows, are sensitively dependent on the trap strength parameter, as well as the initial displacement of the kink and antikink.