∂ -problem for the focusing nonlinear Schrödinger equation and soliton shielding

We consider soliton gas solutions of the focusing nonlinear Schrödinger (NLS) equation, where the point spectrum of the Zakharov-Shabat linear operator condenses in a bounded domain D in the upper half-plane. We show that the corresponding inverse scattering problem can be formulated as a ∂-problem on the complex plane. We prove that the function of the N soliton solution converges in the limit N→∞ to the function (a Fredholm determinant) of the ∂-problem. Furthermore, we prove that such a function is non-vanishing for all values of x and t, thus showing the existence of a solution of the ∂-problem. Then we show that, when the domain D is an ellipse and the soliton gas spectral data are analytic, the inverse problem reduces to the soliton spectra concentrating on the segment connecting the foci of the ellipse (soliton shielding). The NLS solution for fixed times is asymptotically step-like oscillatory, and it is described by a periodic elliptic function as x→-∞ while it vanishes exponentially fast as x→+∞.