On a kinetic equation in weak turbulence theory for the nonlinear Schr\"odinger equation

TL;DR

This paper reviews and contextualizes previous results on a quadratic kinetic equation related to weak turbulence in nonlinear Schrödinger equations, and explores conjectures on self-similar solutions supported by analysis and numerics.

Contribution

It summarizes key results in weak turbulence theory for nonlinear Schrödinger equations and introduces conjectures on self-similar solutions with supporting analysis and numerical evidence.

Findings

Summary of long-time asymptotics in weak turbulence theory

Presentation of two conjectures on self-similar solutions

Numerical and consistency analysis supporting conjectures

Abstract

The results from J. Stat. Phys. 159:668-712 & 163:1350-1393, on a quadratic kinetic equation in the analysis of the long time asymptotics of weak turbulence theory for the nonlinear Schr\"odinger equation, are summarized and placed in context. Additionally, two conjectures on self-similar solutions are presented, and backed with consistency analysis and numerics.