Exploring Breather Dynamics in a Two-dimensional, Nonlinear Schroedinger Equation with Non-local Derivatives

Breather solutions to nonlinear wave equations represent an important class of coherent structure, typically featuring strong spatial localization and oscillations in time. In our talk, we present a study of the nonlinear Schroedinger equation (NLS) with non-local derivatives evaluated on a periodic, two-dimensional domain. For derivatives of certain orders, we find a novel breather solution that dominates field evolution in the regime of nonlinearity approaching zero. As nonlinearity is increased, the breathers break down, yielding to the wave-turbulence (or Rayleigh-Jeans) spectra. To better understand these dynamics, we study the phase-space trajectories associated with the breather solutions and find that they are quasi-periodic and close to trajectories of the linearized NLS. With the increase of nonlinearity, these trajectories deform before breaking down entirely, revealing a connection between the breather solution and Kolmogorov-Arnold-Moser (KAM) theory. We conclude by exploring this connection briefly, raising questions for future work.