The sine-Gordon equation in the semiclassical limit: dynamics of fluxon condensates

TL;DR

This paper analyzes the semiclassical limit of the sine-Gordon equation with impulse data, deriving explicit solutions involving elliptic functions and confirming their consistency with Whitham's modulation theory.

Contribution

It provides explicit elliptic function solutions for the sine-Gordon equation in the semiclassical regime and validates them against Whitham's modulation theory predictions.

Findings

Explicit formulas describe high-frequency fluxon dynamics.

Results align with Whitham's theory in both stable and unstable regimes.

The approach captures initial high-frequency motions accurately.

Abstract

We study the Cauchy problem for the sine-Gordon equation in the semiclassical limit with pure-impulse initial data of sufficient strength to generate both high-frequency rotational motion near the peak of the impulse profile and also high-frequency librational motion in the tails. We show that for small times independent of the semiclassical scaling parameter, both types of motion are accurately described by explicit formulae involving elliptic functions. These formulae demonstrate consistency with predictions of Whitham's formal modulation theory in both the hyperbolic (modulationally stable) and elliptic (modulationally unstable) cases.