Kink-antikink interaction forces and bound states in a φ 4 model with quadratic and quartic dispersion

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.

Automated summary

The study investigates the interaction of solitary waves in the well-known phi (4) Klein-Gordon theory, identifying three distinct cases based on the competition of linear operators with different effects. Each case results in a different form of inter-wave interaction, including oscillatory, exponential, and linearly modulated exponential effects. The acceleration as a function of separation distance is calculated and tested to confirm agreement with predictions from ordinary and partial differential equations, while explanations for disparities are provided.