On self-similar solutions to a kinetic equation arising in weak turbulence theory for the nonlinear Schr\"odinger equation

TL;DR

This paper constructs self-similar solutions with fat tails for a kinetic equation related to weak turbulence in the nonlinear Schrödinger equation, revealing detailed tail behavior and extending previous solutions with finite mass and energy.

Contribution

It introduces a new family of self-similar solutions with finite mass but infinite energy, providing bounds on their tail behavior, advancing understanding of long-term dynamics in weak turbulence.

Findings

Constructed self-similar solutions with fat tails.

Proved exponential bounds on the tails of solutions.

Extended previous solutions with finite mass and energy.

Abstract

We construct a family of self-similar solutions with fat tails to a quadratic kinetic equation. This equation describes the long time behaviour of weak solutions with finite mass to the weak turbulence equation associated to the nonlinear Schr\"odinger equation. The solutions that we construct have finite mass, but infinite energy. In J. Stat. Phys. 159:668-712, self-similar solutions with finite mass and energy were constructed. Here we prove upper and lower exponential bounds on the tails of these solutions.