∂¯-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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="script">D</mml:mi> </mml:mrow> </mml:math> in the upper half-plane. We show that the corresponding inverse scattering problem can be formulated as a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mo accent="false">¯</mml:mo> </mml:mover> </mml:math> -problem on the complex plane. We prove that the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> -function of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> soliton solution converges in the limit <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:math> to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> -function (a Fredholm determinant) of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mo accent="false">¯</mml:mo> </mml:mover> </mml:math> -problem. Furthermore, we prove that such a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> -function is non-vanishing for all values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>x</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>t</mml:mi> </mml:math> , thus showing the existence of a solution of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mo accent="false">¯</mml:mo> </mml:mover> </mml:math> -problem. Then we show that, when the domain <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="script">D</mml:mi> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>x</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mo>−</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:math> while it vanishes exponentially fast as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>x</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mo>+</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:math> .