Dark soliton gas and the generalized Riemann problem for the defocusing nonlinear Schrödinger equation
by
B/2nd floor-B.203 - Seminar room
Marc de Hemptinne (chemin du Cyclotron, 2, Louvain-la-Neuve)
We discuss dark soliton gas and its connection with the generalized Riemann problem for the defocusing nonlinear Schrödinger equation. The latter extends the classical problem by replacing its constant states with periodic finite-gap backgrounds. Starting from a continuum limit of multisoliton solutions, we construct arbitrary-genus gas potentials that approach a finite-gap wave on the left and a constant background on the right. For step-like initial data approaching two elliptic backgrounds, we develop direct scattering and transform the inverse problem into a full dark soliton gas Riemann–Hilbert problem with a radiative component. Under suitable spectral and regularity assumptions, we analyze the long-time behavior of the resulting solutions using the Deift–Zhou nonlinear steepest descent method. This talk is based on three related works. The first is joint work with Marco Bertola, Dengshan Wang, and Peng Yan, and the third is joint work with Dengshan Wang.