Collective Coordinates in One-Dimensional Soliton Models Revisited

TL;DR

This paper evaluates the effectiveness of collective coordinate methods in modeling kink-antikink interactions in one-dimensional scalar field theories, focusing on vibrational modes' role in resonance phenomena.

Contribution

It compares numerical solutions with simplified collective coordinate models to assess their validity in capturing resonance structures in kink-antikink scattering.

Findings

Collective coordinate approximation partially captures resonance effects.

Vibrational modes influence resonance structures in kink-antikink interactions.

Simplified models provide insights but have limitations compared to full solutions.

Abstract

We compare numerical solutions to the full field equations to simplified approaches based on implementing three collective coordinates for kink-antikink interactions within the $\varphi^4$ and $\phi^6$ models in one time and one space dimensions. We particularly pursue the question whether the collective coordinate approximation substantiates the conjecture that vibrational modes are important for resonance structures to occur in kink-antikink scattering.