Inhomogeneous turbulence for the Wick nonlinear Schr\"odinger equation

TL;DR

This paper introduces the Wick NLS model, a simplified wave turbulence model without self-interactions, deriving new wave kinetic equations for both homogeneous and inhomogeneous regimes that describe the statistical behavior of solutions.

Contribution

It presents a novel simplified wave turbulence model (Wick NLS) and derives new wave kinetic equations for inhomogeneous settings, including a refined collision description.

Findings

Derived wave kinetic equations for homogeneous and inhomogeneous regimes.

Identified a new wave kinetic equation capturing refined collision dynamics.

Established the model's consistency with formal predictions in the homogeneous case.

Abstract

We introduce a simplified model for wave turbulence theory -- the Wick NLS, of which the main feature is the absence of all self-interactions in the correlation expansions of its solutions. For this model, we derive several wave kinetic equations that govern the effective statistical behavior of its solutions in various regimes. In the homogeneous setting, where the initial correlation is translation invariant, we obtain a wave kinetic equation similar to the one predicted by the formal theory. In the inhomogeneous setting, we obtain a wave kinetic equation that describes the statistical behavior of the wavepackets of the solutions, accounting for both the transport of wavepackets and collisions among them. Another wave kinetic equation, which seems new in the literature, also appears in a certain scaling regime of this setting and provides a more refined collision picture.