D.E. Pelinovsky and R.E. White
Localized structures on librational and rotational travelling waves in the sine-Gordon equation
Proc. R. Soc. A 476: 20200490 (18 pages)
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
uxon condensates in the semi-classical
limit. The different dynamics are related to
modulational instability of the librational waves and
modulational stability of the rotational waves.
Keywords:
sine-Gordon equation, travelling waves, modulational instability, rogue waves, semiclassical dynamcis.