Spontaneous emergence of two-dimensional quasibreathers in a nonlinear Schrödinger equation with nonlocal derivatives

We consider the nonlinear Schr\"odinger equation with nonlocal derivatives in a two-dimensional periodic domain. For certain orders of derivatives, we find a type of quasi-breather solution dominating the field evolution at low nonlinearity levels. With the increase of nonlinearity, the structures break down, giving way to Rayleigh-Jeans (or wave turbulence) spectra. Phase-space trajectories associated with the quasibreather solutions are found to be close to that of the linear system and almost periodic. We employ two methods to search for nearby periodic solutions (e.g., exact breathers), yet none are found. Given these distinguishing behaviors, we interpret this structure in the context of Kolmogorov-Arnold-Moser (KAM) theory.