Breather stripes and radial breathers of the two-dimensional sine-Gordon equation

Abstract

We revisit the problem of transverse instability of a 2D breather stripe of the sine-Gordon (sG) equation. A numerically computed Floquet spectrum of the stripe is compared to ana- lytical predictions developed by means of multiple-scale perturbation theory showing good agreement in the long-wavelength limit. By means of direct simulations, it is found that the instability leads to a breakup of the quasi-1D breather in a chain of interacting 2D radial breathers that appear to be fairly robust in the dynamics. The stability and dy- namics of radial breathers in a finite domain are studied in detail by means of numerical methods. Different families of such solutions are identified. They develop small-amplitude spatially oscillating tails (“nanoptera”) through a resonance of higher-order breather’s har- monics with linear modes (“phonons”) belonging to the continuous spectrum. These re- sults demonstrate the ability of the 2D sG model within our finite domain computations to localize energy in long-lived, self-trapped breathing excitations.