Law of Large Numbers and Central Limit Theorem for random sets of solitons of the focusing nonlinear Schrödinger equation

We study a random configuration of $N$ soliton solutions $ψ_N(x,t;\boldsymbolλ)$ of the cubic focusing Nonlinear Schrödinger (fNLS) equation in one space dimension. The $N$ soliton solutions are parametrized by $2N$ complex numbers $(\boldsymbolλ, \boldsymbol{c})$ where $\boldsymbolλ\in\mathbb{C}_+^N$ are the eigenvalues of the Zakharov-Shabat linear operator, and $ \boldsymbol{c}\in\mathbb{C}^N\backslash \{0\}$ are the norming constants of the corresponding eigenfunctions. The randomness is obtained by choosing the complex eigenvalues to be i.i.d. random variables sampled from a probability distribution with compact support in the complex plane. The corresponding norming constants are interpolated by a smooth function of the eigenvalues. Then we consider the expectation of the random measure associated to this random spectral data. Such expectation uniquely identifies, via the Zakharov-Shabat inverse spectral problem, a solution $ψ_\infty(x,t)$ of the fNLS equation. This solution can be interpreted as a soliton gas solution. We prove a Law of Large Numbers and a Central Limit Theorem for the differences $ψ_N(x,t;\boldsymbolλ)-ψ_\infty(x,t)$ and $|ψ_N(x,t;\boldsymbolλ)|^2-|ψ_\infty(x,t)|^2$ when $(x,t)$ are in a compact set of $\mathbb R\times\mathbb R^+$; we additionally compute the correlation functions.