Kink–antikink interaction forces and bound states in a biharmonic ϕ ⁴ model

Abstract

We consider the interaction of solitons in a biharmonic, beam model analogue of the well-studied φ 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 ordinary differential equations. 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 partial differential equations 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.

Chains of Interacting Solitons

“…However, the situation can be different if we extend the model (1), for example, we can consider coupled two-component system with one of the scalar components having the kink structure and the second component being a non-topological soliton [30,31], or modify the model ( 1), in such a way, that it still supports the kinks but possess a biharmonic spatial derivative term [32]. In the latter case, it is possible to construct a static kink-anti-kink pair.…”

Chains of interacting solitons

“…In other words, there is no static multisoliton solutions in the model (1). However, the situation can be different if we extend the model (1), for example we can consider coupled twocomponent system with one of the scalar components having the kink structure and the second component being a nontopological soliton [30,31], or modify the model (1), in such a way, that it still supports the kinks but possess a biharmonic spatial derivative term [32]. In the latter case it is possible to construct a static kink-antikink pair.…”

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

“…In [30,33], a variant of this equation was explored where the harmonic spatial derivative term was replaced by a biharmonic term of the form:…”