Solitons in the Presence of a Small, Slowly Varying Electric Field

TL;DR

This paper analyzes how solitons in the sine-Gordon equation are affected by a small, slowly varying electric field, showing that solutions closely follow a perturbed solitary wave trajectory over long times.

Contribution

It demonstrates the persistence and approximate dynamics of sine-Gordon solitons under small, slowly varying external electric fields, with precise error estimates and reduced ODE models.

Findings

Solutions stay close to the solitary manifold for time up to 1/ε

Trajectory on the solitary manifold is governed by approximate Hamiltonian ODEs

Error bounds of order ε^{3/4} and ε^3 for solutions and trajectories

Abstract

We consider the perturbed sine-Gordon equation $\theta_{tt}-\theta_{xx}+\sin \theta= \varepsilon^2 f(\varepsilon x)$, where the external perturbation $\varepsilon^2 f(\varepsilon x)$ corresponds to a small, slowly varying electric field. We show that the initial value problem with an appropriate initial state close enough to the solitary manifold has a unique solution, which follows up to time $1/{\varepsilon}$ and errors of order $\varepsilon^{ 3/4}$ a trajectory on the solitary manifold. The trajectory on the solitary manifold is described by ODEs, which agree approximately up to errors of order $\varepsilon^3$ with Hamilton equations for the restricted to the solitary manifold sine-Gordon Hamiltonian.