Homogenization for the nonlinear Schrödinger equation with sprinkled nonlinearity

1. Problem Statement and Motivation

The paper investigates the homogenization of the Nonlinear Schrödinger Equation (NLS) where the nonlinear interaction is not distributed continuously (as in the standard cubic NLS) but is instead concentrated at a random set of points ("sprinkled") throughout the spatial domain.

• The Model: The equation is given by:
  $$i\partial_t \psi = -\partial_x^2 \psi + 2|\psi|^2 \psi \, d\mu$$
  where $\mu$ is a random measure representing the distribution of defects. In the standard case, $\mu$ is the Lebesgue measure ($dx$), yielding the usual cubic NLS. In the "sprinkled" case, $\mu$ is modeled as a homogeneous Poisson process of unit intensity (or more generally, a stationary random measure with independent increments and unit intensity).
• Physical Context: This models wave propagation in media where nonlinear effects (like the optical Kerr effect) occur only at specific, randomly distributed defects.
• The Homogenization Limit: The authors consider a sequence of equations $(NLS_n)$ where the defects become increasingly dense. Specifically, they rescale the measure $\mu$ by a parameter $\epsilon_n \to 0$. The goal is to prove that as the distance between defects becomes much smaller than the wavelength, the solution $\psi_n$ converges to the solution $\psi$ of the standard homogeneous cubic NLS:
  $$i\partial_t \psi = -\partial_x^2 \psi + 2|\psi|^2 \psi$$
• The Fluctuation Problem: Beyond simple convergence, the paper asks how the solutions $\psi_n$ fluctuate around the homogenized limit $\psi$. By formally expanding $\psi_n = \psi + \sqrt{\epsilon_n} \phi_n$, the authors investigate the limiting behavior of the fluctuation term $\phi_n$.

2. Methodology

The authors employ a combination of probabilistic analysis, harmonic analysis, and nonlinear PDE techniques.

A. Probabilistic Framework

• Random Measures: The randomness is encoded in the measure $\mu_n$. The authors utilize the Laplace functional of the random measure to characterize its distribution. They assume the measure has independent increments and unit intensity, with a Lévy measure $\Lambda$ satisfying specific integrability conditions (ensuring analyticity of the cumulant generating function).
• White Noise Limit: A crucial step is showing that the rescaled fluctuation of the measure, $\nu_n = \frac{1}{\sqrt{\epsilon_n}}(\mu_n - dx)$, converges in distribution to spatial white noise $\xi$. This allows the fluctuation equation to be treated as a linear PDE driven by white noise.

B. Functional Spaces and Weights

To handle the singular nature of the measure $\mu_n$ (which is not a function but a distribution), the authors introduce specialized weighted spaces:

• Weighted $L^2$ Spaces ($L^2_Z$): They define a weight $\omega(x; Z)$ based on a sequence $Z$ derived from the local mass of the measure $\mu_n$ on dyadic intervals.
• Solution Spaces ($X^1_n, Y^1_{n,s}$): They construct Banach spaces $X^1_n$ and $Y^1_{n,s}$ equipped with norms that incorporate these weights. These spaces are tailored to control the interaction between the solution and the singular measure.
• Key Technical Tool: They prove that for initial data in $H^1$, the solution almost surely belongs to these weighted spaces uniformly in $n$.

C. Estimation Techniques

• Moment-Cumulant Formula: To control the moments of the random variables arising from the interaction with $\mu_n$, the authors use the joint cumulant expansion. They leverage the independence of increments of the Poisson process to show that higher-order cumulants decay sufficiently fast.
• Orthonormal Basis: They utilize the Haar basis to decompose the random measure fluctuations. This allows them to treat the random terms as sums of orthonormal random variables, facilitating moment estimates.
• Gronwall and Radon-Riesz: Standard PDE techniques (Gronwall's inequality for time evolution, Radon-Riesz theorem for upgrading weak convergence to strong convergence in $H^1$) are adapted to the probabilistic setting.

3. Key Contributions and Results

The paper establishes two main theorems:

Theorem 1.1: Homogenization (Strong Convergence)

• Result: As $\epsilon_n \to 0$, the solution $\psi_n$ of the sprinkled NLS converges to the solution $\psi$ of the standard cubic NLS in the space $L^p(\Omega; C([-T, T]; H^1))$.
• Significance: This rigorously justifies the physical intuition that a "randomly sprinkled" nonlinear medium behaves like a homogeneous medium on large scales, provided the defects are dense enough. The convergence is in the strong $H^1$ topology, which is non-trivial given the singular nature of the nonlinearity.

Theorem 1.2: Fluctuations (Central Limit Theorem)

• Result: The rescaled fluctuation $\phi_n = \frac{1}{\sqrt{\epsilon_n}}(\psi_n - \psi)$ converges in distribution (as a random variable in $C([-T, T]; H^{-s})$ for $s > 1/2$) to a solution $\phi$ of a linearized stochastic NLS:
  $$i\partial_t \phi = -\partial_x^2 \phi + 4|\psi|^2 \phi + 2\psi^2 \bar{\phi} + 2|\psi|^2 \psi \xi$$
  where $\xi$ is spatial white noise.
• Significance:
  • This establishes a Central Limit Theorem for the NLS with random nonlinearity.
  • The limit $\phi$ is a Gaussian process with mean zero.
  • The covariance structure is explicitly determined by the linearization of the deterministic NLS around the solution $\psi$.
  • This is the first result of its kind for the Schrödinger equation with inhomogeneity arising from nonlinear interactions (as opposed to linear potentials).

Appendix B: Mollified Measures

The authors extend their results to a "mollified" version of the model where the Dirac-like defects are smoothed out by a kernel $\zeta_h$. They prove that the homogenization and fluctuation results hold uniformly as the smoothing parameter $h \to 0$, bridging the gap between the singular measure model and physically realistic smooth potentials.

4. Significance and Impact

1. Universality of Homogenization: The results demonstrate that the homogenized limit (standard NLS) and the fluctuation limit (linearized NLS with white noise) are universal. They depend only on the statistical properties of the random measure (stationarity, independent increments, unit intensity) and not on the specific microscopic arrangement (e.g., Poisson vs. other processes).
2. Handling Singular Nonlinearities: The paper develops a robust framework for analyzing NLS equations where the nonlinearity is a distribution (measure) rather than a function. This is a significant technical advancement over previous work which often required bounded or periodic coefficients.
3. Central Limit Theorem for Nonlinear PDEs: By characterizing the fluctuations as a Gaussian process driven by white noise, the paper provides a statistical description of the error in homogenization. This is crucial for understanding the stability and predictability of wave propagation in disordered media.
4. Comparison to Linear Cases: While homogenization for linear elliptic and parabolic equations is well-studied, the nonlinear Schrödinger case presents unique challenges due to the lack of maximum principles and the specific structure of the nonlinearity. This work fills a gap in the literature regarding stochastic homogenization for dispersive nonlinear equations.

In summary, the paper provides a rigorous mathematical foundation for understanding how random, concentrated nonlinearities average out to produce standard wave dynamics, while precisely quantifying the random deviations from this average behavior.