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

We consider the nonlinear Schrödinger 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.

Effect of discrete resonant manifold structure on discrete wave turbulence.

We consider the long-term dynamics of nonlinear dispersive waves in a finite periodic domain. The purpose of the work is to show that the statistical properties of the wave field rely critically on the structure of the discrete resonant manifold (DRM). To demonstrate this, we simulate the two-dimensional Majda-McLaughlin-Tabak equation on rational and irrational tori, resulting in remarkably different power-law spectra and energy cascades at low nonlinearity levels. The difference is explained in terms of different structures of the DRM, which makes use of recent number theory results.