Homogenization for the nonlinear Schrödinger equation with sprinkled nonlinearity

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.

💡 Research Summary

The paper investigates the large‑scale behavior of a one‑dimensional nonlinear Schrödinger equation (NLS) whose nonlinearity is “sprinkled’’ at random points in space. The random set of points is modeled by a stationary random measure µ with independent increments and unit intensity; in the canonical case µ is a homogeneous Poisson process. The equation under study is

  (NLSµ)  i∂ₜψ = –∂ₓ²ψ + 2|ψ|²ψ dµ,

where the measure µ is interpreted as a Schwartz distribution, so the cubic nonlinearity is multiplied by the random measure. The authors consider a family of rescaled random measures µₙ defined through the Laplace functional

  E exp(–∫f dµₙ) = exp(–εₙ⁻¹∫Φ(εₙf) dx + εₙ∫f dx),

with εₙ→0 and Φ(z)=∫₀^∞(1–sz–e^{–sz}) dΛ(s) determined by the Lévy measure Λ of µ. The scaling is chosen so that the random field εₙ⁻¹µₙ(εₙ·) has the same law as µ. The corresponding NLSₙ equation is

  (NLSₙ)  i∂ₜψₙ = –∂ₓ²ψₙ + 2|ψₙ|²ψₙ dµₙ.

The first main result (Theorem 1.1) proves homogenization: for any fixed H¹ initial datum ψ₀, the solution ψₙ of (NLSₙ) converges in probability, uniformly on compact time intervals, to the solution ψ of the standard cubic NLS

  (NLS)  i∂ₜψ = –∂ₓ²ψ + 2|ψ|²ψ,

in the sense

  E sup_{|t|≤T}‖ψₙ(t)–ψ(t)‖_{H¹}^p → 0 as n→∞

for every T>0 and 1≤p<∞. The proof follows a Duhamel expansion of the difference ψₙ–ψ, estimates the nonlinear difference term in L^∞ and H^{–1}, and then uses conservation of mass and energy to control the H¹ norm. A crucial technical device is the introduction of weighted function spaces X₁ⁿ and Y₁ⁿ,ₛ that incorporate the random local mass of µₙ (the quantities Zₙ(k)=µₙ(