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Homogenization for the nonlinear Schrödinger equation with sprinkled nonlinearity

Benjamin Harrop-Griffiths, Maria Ntekoume

TL;DR

The paper analyzes a nonlinear Schrödinger equation with sprinkled nonlinearity distributed by a Poisson random measure. It proves rigorous homogenization to the cubic NLS and derives a central-limit-type fluctuation equation for the deviation, driven by white noise. The approach combines Laplace-functionals of random measures, Haar-based moment bounds, and robust a priori estimates to control randomness and obtain distributional convergence in $C([-T,T];H^{-s})$ for $s>1/2$. The results extend stochastic homogenization in nonlinear dispersive PDEs beyond periodic or bounded-coupling settings and reveal a universal limiting structure for a broad class of random measures.

Abstract

We first prove homogenization for the nonlinear Schrödinger equation with sprinkled nonlinearity introduced in [19]. We then investigate how solutions fluctuate about the homogenized solution.

Homogenization for the nonlinear Schrödinger equation with sprinkled nonlinearity

TL;DR

The paper analyzes a nonlinear Schrödinger equation with sprinkled nonlinearity distributed by a Poisson random measure. It proves rigorous homogenization to the cubic NLS and derives a central-limit-type fluctuation equation for the deviation, driven by white noise. The approach combines Laplace-functionals of random measures, Haar-based moment bounds, and robust a priori estimates to control randomness and obtain distributional convergence in for . The results extend stochastic homogenization in nonlinear dispersive PDEs beyond periodic or bounded-coupling settings and reveal a universal limiting structure for a broad class of random measures.

Abstract

We first prove homogenization for the nonlinear Schrödinger equation with sprinkled nonlinearity introduced in [19]. We then investigate how solutions fluctuate about the homogenized solution.
Paper Structure (5 sections, 21 theorems, 236 equations)

This paper contains 5 sections, 21 theorems, 236 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Homogenization
  4. Fluctuations
  5. Solutions of \ref{['linear toy']}

Key Result

Theorem 1.1

Let $0<\varepsilon_n\to0$ and $\psi_0\in H^1$. If $\psi_n\in C(\mathbb{R};H^1)$ is the almost surely defined solution of NLSn with initial data $\psi_n(0) = \psi_0$ and $\psi\in C(\mathbb{R};H^1)$ is the solution of NLS with initial data $\psi(0) = \psi_0$ then, for all $T>0$ and $1\leq p<\infty$, w

Theorems & Definitions (42)

  • Theorem 1.1: Homogenization
  • Proposition 1.2
  • Theorem 1.3: Fluctuations
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 32 more