Kink-Antikink Interaction Forces and Bound States in a $\phi^4$ Model with Quadratic and Quartic dispersion

TL;DR

This paper investigates how quadratic and quartic dispersion terms in a $^4$ Klein-Gordon model influence solitary wave interactions, revealing different regimes of force behavior and collision dynamics.

Contribution

It introduces a comprehensive analysis of kink-antikink interactions in a $^4$ model with mixed dispersion effects, identifying distinct regimes and their impact on forces and collision outcomes.

Findings

Three regimes of interaction depending on model parameters.

Good agreement between ODE and PDE predictions for forces.

Initial insights into collision dynamics with mixed dispersion effects.

Abstract

We consider the interaction of solitary waves in a model involving the well-known $\phi^4$ Klein-Gordon theory, but now bearing both Laplacian and biharmonic terms with different prefactors. As a result of the competition of the respective linear operators, we obtain three distinct cases as we vary the model parameters. In the first the biharmonic effect dominates, yielding an oscillatory inter-wave interaction; in the third the harmonic effect prevails yielding exponential interactions, while we find an intriguing linearly modulated exponential effect in the critical second case, separating the above two regimes. For each case, we calculate the force between the kink and antikink when initially separated with sufficient distance. Being able to write the acceleration as a function of the separation distance, and its corresponding ordinary differential equation, we test the corresponding predictions, finding very good agreement, where appropriate, with the corresponding partial differential equation results. Where the two findings differ, we explain the source of disparities. Finally, we offer a first glimpse of the interplay of harmonic and biharmonic effects on the results of kink-antikink collisions and the corresponding single- and multi-bounce windows.