Localized structures on librational and rotational travelling waves in the sine-Gordon equation

TL;DR

This paper derives exact solutions to the sine-Gordon equation that describe localized structures on librational and rotational travelling waves, revealing phenomena like rogue waves and kinks with algebraic decay, relevant to fluxon condensates.

Contribution

It introduces new exact solutions for the sine-Gordon equation modeling localized structures on different wave backgrounds, highlighting their dynamics and stability.

Findings

Librational waves can host rogue wave solutions with algebraic decay.

Rotational waves support kink solutions propagating on periodic backgrounds.

The solutions are relevant for understanding fluxon condensate dynamics.

Abstract

We derive exact solutions to the sine--Gordon equation describing localized structures on the background of librational and rotational travelling waves. In the case of librational waves, the exact solution represents a localized spike in space-time coordinates (a rogue wave) which decays to the periodic background algebraically fast. In the case of rotational waves, the exact solution represents a kink propagating on the periodic background and decaying algebraically in the transverse direction to its propagation. These solutions model the universal patterns in the dynamics of fluxon condensates in the semi-classical limit. The different dynamics is related to different outcomes of modulational stability of the librational and rotational waves.