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

•We studied the transverse instability of quasi-1D breather stripes of the 2D sine-Gordon equation.•We developed a variational approximation capturing the necking of quasi-1D breather stripes.•We observed that the breather stripe instability leads to a chain of interacting 2D radial breathers.•We examined the potential existence and stability of radial breathers in finite domains. 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 analytical 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 dynamics 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 harmonics with linear modes (“phonons”) belonging to the continuous spectrum. These results demonstrate the ability of the 2D sG model within our finite domain computations to localize energy in long-lived, self-trapped breathing excitations.