Wobbling kinks in $\phi^4$ theory

TL;DR

This paper develops a detailed asymptotic analysis of wobbling kinks in  theory, deriving their long-term behavior and confirming the decay law of wobbling amplitude through matched asymptotics and nonlinear damping equations.

Contribution

It provides a uniform asymptotic expansion for wobbling kinks to all orders, describing radiation and amplitude decay with a new analytical approach.

Findings

Wobbling kink amplitude decays as t^{-1/2} over time.

Long-range radiation behavior is characterized by matched asymptotic expansions.

The amplitude dynamics follow a simple nonlinear damping ODE.

Abstract

We present a uniform asymptotic expansion of the wobbling kink to any order in the amplitude of the wobbling mode. The long-range behaviour of the radiation is described by matching the asymptotic expansions in the far field and near the core of the kink. The complex amplitude of the wobbling mode is shown to obey a simple ordinary differential equation with nonlinear damping. We confirm the $t^{-1/2}$-decay law for the amplitude which was previously obtained on the basis of energy considerations.