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 ψN(x,t;λ) of the cubic focusing nonlinear Schrödinger (fNLS) equation in one space dimension. The N soliton solutions are parametrized by 2N complex numbers (λ,c), where λ∈C+N are the eigenvalues of the Zakharov–Shabat linear operator, and c∈CN∖{0} are the norming constants of the corresponding eigenfunctions. The randomness is obtained by choosing the complex eigenvalues to be independent and identically distributed 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∞(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 ψN(x,t;λ)−ψ∞(x,t) and |ψN(x,t;λ)|2−|ψ∞(x,t)|2 when (x,t) are in a compact set of R×R+; we additionally compute the correlation functions.