Deformation of sine-Gordon two-soliton solutions in ${\phi }^4$ kink-antikink configurations

Abstract

In this work, we construct analytical kink-antikink $(K\bar{K})$ configurations in the non-integrable ${\phi }^4$ model by mapping exact solutions of the integrable sine-Gordon system via a field deformation. This procedure yields two distinct classes of configurations, parametrized by the initial half-separation and velocity. We compare these profiles with the standard additive superposition Ansatz in terms of vacuum structure and equation-of-motion residuals, deriving explicit closed-form expressions for the corresponding integrated squared residuals. For small separations, the mapped soliton-antisoliton configurations exhibit appreciably smaller residuals than the naive superposition Ansatz, which in turn performs better than the mapped two-soliton configurations. Although the mapped fields are not exact solutions of the ${\phi }^4$ equation of motion, they provide mathematically consistent, topologically sound, and physically motivated initial data for numerical studies of kink-antikink scattering and resonance phenomena.

9 pages, 4 figure