Kink-Antikink Interaction Forces and Bound States in a Biharmonic $\phi^4$ Model

TL;DR

This paper investigates the interaction forces between kink and antikink solitons in a biharmonic $^4$ model, revealing oscillatory forces, bound states, and collision dynamics through analytical and numerical methods.

Contribution

It introduces a novel analysis of kink-antikink interactions in a biharmonic $^4$ model, highlighting oscillatory forces and bound states not seen in standard models.

Findings

Force exhibits exponentially-decaying oscillatory behavior

Infinite set of kink-antikink bound states predicted and confirmed

Collision dynamics influenced by the oscillatory interaction force

Abstract

We consider the interaction of solitons in a biharmonic, beam model analogue of the well-studied $\phi^4$ Klein-Gordon theory. Specifically, we calculate the force between a well separated kink and antikink. Knowing their accelerations as a function of separation, we can determine their motion using a simple ODE. There is good agreement between this asymptotic analysis and numerical computation. Importantly, we find the force has an exponentially-decaying oscillatory behaviour (unlike the monotonically attractive interaction in the Klein-Gordon case). Corresponding to the zeros of the force, we predict the existence of an infinite set of field theory equilibria, i.e., kink-antikink bound states. We confirm the first few of these at the PDE level, and verify their anticipated stability or instability. We also explore the implications of this interaction force in the collision between a kink and an oppositely moving antikink.