2024/09/23 by Marco Bertola, Bertola, Marco, Тамара Грава +3
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Waves and Solitons #Quantum optics and atomic interactions
paper · pdf · doi:10.48550/arxiv.2409.14825
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 D in the upper half-plane. We show that the corresponding inverse scattering problem can be formulated as a ∂-problem on the domain. We prove the existence of the solution of this ∂-problem by showing that the τ-function of the problem (a Fredholm determinant) does not vanish. We then represent the solution of the NLS equation via the τ of the ∂- problem. Finally we show that, when the domain 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 → - ∞ while it vanishes exponentially fast as x → +∞.