$\bar{\partial }$-problem for focusing nonlinear Schrödinger equation and soliton shielding

Abstract

We consider soliton gas solutions of the Focusing Nonlinear Schrödinger (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 $\ddot{I}„$-function of the problem (a Fredholm determinant) does not vanish. We then represent the solution of the NLS equation via the $\ddot{I}„$ 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 $x\to +\infty$.

24 pages, 4 figures