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

Abstract

We consider the perturbed sine-Gordon equation ${\theta }_{tt}-{\theta }_{xx}+\sin \theta ={\epsilon }^{2}f(\epsilon x)$, where the external perturbation ${\epsilon }^{2}f(\epsilon x)$ 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 $1/\epsilon$ and errors of order ${\epsilon }^{3/4}$ a trajectory on the solitary manifold. The trajectory on the solitary manifold is described by ODEs, which agree approximately up to errors of order ${\epsilon }^3$ with Hamilton equations for the restricted to the solitary manifold sine-Gordon Hamiltonian.

arXiv admin note: text overlap with arXiv:1705.05713

References in corpus (1)

• Solitary Wave Dynamics in an External Potential

Cited by in corpus (1)

• Invariant Virtual Solitary Manifold of the Perturbed Sine-Gordon Equation