$\bar{\partial}$-problem for focusing nonlinear Schr\"odinger equation and soliton shielding

TL;DR

This paper develops a $ar{ ext{d}}$-problem approach to soliton gas solutions of the focusing NLS equation, revealing asymptotic step-like oscillatory behavior and connecting inverse scattering with elliptic functions.

Contribution

It formulates the inverse scattering problem as a $ar{ ext{d}}$-problem, proves existence of solutions via Fredholm determinants, and characterizes asymptotic behavior for elliptic domains.

Findings

Inverse scattering as a $ar{ ext{d}}$-problem for soliton gases

Existence of solutions via non-vanishing $ au$-function

Asymptotic step-like oscillatory initial data for elliptic domains

Abstract

We consider soliton gas solutions of the Focusing Nonlinear Schr\"odinger (NLS) equation, where the point spectrum of the Zakharov-Shabat linear operator condensate in a bounded domain $\mathcal{D}$ in the upper half-plane. We show that the corresponding inverse scattering problem can be formulated as a $\overline{\partial}$-problem on the domain. We prove the existence of the solution of this $\overline{\partial}$-problem by showing that the $\tau$-function of the problem (a Fredholm determinant) does not vanish. We then represent the solution of the NLS equation via the $\tau$ of the $\overline{\partial}$- problem. Finally we show that, when the domain $\mathcal{D}$ is an ellipse and the density of solitons is analytic, the initial datum of the Cauchy problem is asymptotically step-like oscillatory, and it is described by a periodic elliptic function as $x \to - \infty$ while it vanishes exponentially fast as $x \to +\infty$.