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The focusing nonlinear Schrödinger equation with step-like oscillating background

Jonatan Lenells
Schedule: 
Monday, January 17, 2022 - 14:00 to 15:00
Abstract: 

I will discuss joint work with Anne Boutet de Monvel and Dmitry Shepelsky where we study the asymptotic behavior of solutions of the nonlinear Schrödinger equation. More precisely, we consider the Cauchy problem for the focusing nonlinear Schrödinger equation with initial data approaching different plane waves at plus and minus infinity. Using Riemann–Hilbert techniques and Deift-Zhou steepest descent arguments, we study the long-time behavior of the solution. We show that there is a wide range of possible asymptotic scenarios. We propose a method for rigorously establishing the existence of certain higher-genus asymptotic sectors, and we compute detailed asymptotic formulas in a genus three sector, i.e., in a sector where the leading term of the asymptotics is given in terms of hyperelliptic functions attached to a Riemann surface of genus three.

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