Breather Solutions to a Two-dimensional Nonlinear Schr\"odinger Equation with Non-local Derivatives

TL;DR

This paper investigates breather solutions in a 2D nonlinear Schrödinger equation with non-local derivatives, revealing their emergence at low nonlinearity and transition to turbulence as nonlinearity increases, with connections to KAM theory.

Contribution

It introduces a new class of breather solutions in a 2D non-local nonlinear Schrödinger equation and links their phase-space behavior to KAM theory.

Findings

Breather solutions dominate at low nonlinearity levels.

Breathers break down into wave turbulence with increased nonlinearity.

Phase-space trajectories of breathers resemble those of the linear system.

Abstract

We consider the nonlinear Schr\"odinger equation with non-local derivatives in a two-dimensional periodic domain. For certain orders of derivatives, we find a new type of breather solution dominating the field evolution at low nonlinearity levels. With the increase of nonlinearity, the breathers break down, giving way to wave turbulence (or Rayleigh-Jeans) spectra. Phase-space trajectories associated with the breather solutions are found to be close to that of the linear system, revealing a connection between the breather solution and Kolmogorov-Arnold-Moser (KAM) theory.