On the Theory of Weak Turbulence for the Nonlinear Schr\"odinger Equation

TL;DR

This paper investigates the kinetic equation in weak turbulence theory for the nonlinear Schrödinger equation, establishing well-posedness, analyzing long-term behavior, and discovering solutions with pulsating dynamics.

Contribution

It introduces new well-posedness results and analyzes qualitative behaviors of solutions, including blow-up, condensation, and pulsating solutions, in the context of weak turbulence theory.

Findings

Proved local and global well-posedness of solutions.

Identified long-time asymptotics and blow-up phenomena.

Discovered solutions exhibiting pulsating behavior.

Abstract

We study the Cauchy problem for a kinetic equation arising in the weak turbulence theory for the cubic nonlinear Schr\"odinger equation. We define suitable concepts of weak and mild solutions and prove local and global well posedness results. Several qualitative properties of the solutions, including long time asymptotics, blow up results and condensation in finite time are obtained. We also prove the existence of a family of solutions that exhibit pulsating behavior.