Breather gas fission from elliptic potentials in self-focusing media

We present an analytical model of integrable turbulence in the focusing nonlinear Schr\"odinger (fNLS) equation, generated by a one-parameter family of finite-band elliptic potentials in the semiclassical limit. We show that the spectrum of these potentials exhibits a thermodynamic band/gap scaling compatible with that of soliton and breather gases depending on the value of the elliptic parameter m of the potential. We then demonstrate that, upon augmenting the potential by a small random noise (which is inevitably present in real physical systems), the solution of the fNLS equation evolves into a fully randomized, spatially homogeneous breather gas, a phenomenon we call breather gas fission. We show that the statistical properties of the breather gas at large times are determined by the spectral density of states generated by the unperturbed initial potential. We analytically compute the kurtosis of the breather gas as a function of the elliptic parameter m, and we show that it is greater than 2 for all non-zero m, implying non-Gaussian statistics. Finally, we verify the theoretical predictions by comparison with direct numerical simulations of the fNLS equation. These results establish a link between semiclassical limits of integrable systems and the statistical characterization of their soliton and breather gases.