Scattering, random phase and wave turbulence

TL;DR

This paper investigates wave turbulence in the 2D cubic Schrödinger equation on the real plane, linking resonant manifold regularity with dispersive effects and analyzing the evolution operator across different regimes for deterministic and random initial data.

Contribution

It introduces a novel approach by studying NLS on the real plane with dispersive effects, using Gaussian truncation to analyze convergence towards the kinetic operator in multiple regimes.

Findings

Identification of two regimes with different convergence behaviors

Explicit calculations of the evolution operator in various time scales

Linking resonant manifold regularity with dispersive properties

Abstract

We start from the remark that in wave turbulence theory, exemplified by the cubic twodimensional Schr{\"o}dinger equation (NLS) on the real plane, the regularity of the resonant manifold is linked with dispersive properties of the equation and thus with scattering phenomena. In contrast with classical analysis starting with a dynamics on a large periodic box, we propose to study NLS set on the real plane using the dispersive effects, by considering the time evolution operator in various time scales for deterministic and random initial data. By considering periodic functions embedded in the whole space by gaussian truncation, this allows explicit calculations and we identify two different regimes where the operators converges towards the kinetic operator but with different form of convergence.