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

Abstract

We study a random configuration of $N$ soliton solutions ${\psi }_N(x,t;\mathbf{\lambda })$ of the cubic focusing Nonlinear Schrödinger (fNLS) equation in one space dimension. The $N$ soliton solutions are parametrized by $2N$ complex numbers $(\mathbf{\lambda },\mathbf{c})$ where $\mathbf{\lambda }\in {\mathbb{C}}_{+}^N$ are the eigenvalues of the Zakharov-Shabat linear operator, and $\mathbf{c}\in {\mathbb{C}}^N\backslash \{0\}$ 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 ${\psi }_{\infty }(x,t)$ 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 ${\psi }_N(x,t;\mathbf{\lambda })-{\psi }_{\infty }(x,t)$ and $|{\psi }_N(x,t;\mathbf{\lambda }){|}^2-|{\psi }_{\infty }(x,t){|}^2$ when $(x,t)$ are in a compact set of $\mathbb{R}\times {\mathbb{R}}^{+}$; we additionally compute the correlation functions.

27 pages, 2 figures