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The defocusing nonlinear Schrödinger equation with step-like oscillatory data

Time: Mon 2021-06-14 14.00

Location: Webinar registration link: https://kth-se.zoom.us/webinar/register/WN_Wy8LLczrQg-5C6iXAa5-Lw, Stockholm (English)

Subject area: Mathematics

Doctoral student: Samuel Fromm , Matematik (Avd.)

Opponent: Professor Alexander Tovbis, University of Central Florida

Supervisor: Universitetslektor Jonatan Lenells, Matematik (Avd.)

Abstract

The thesis at hand consists of three papers as well as an introductory chapter and a summary of results. The topic of the thesis is the study of the defocusing nonlinear Schrödinger equation with step-like oscillatory data.

Paper A studies the Cauchy problem for the defocusing nonlinear Schrödinger equation on the line with step-like oscillatory boundary conditions. More precisely, the solution is required to approach a single exponential as x → -∞ and to decay to zero as x → +∞. We prove existence of a global solution and show that the solution can be expressed in terms of the solution of a Riemann-Hilbert problem. We also compute the long-time asymptotics of the solution and apply the results to a related initial-boundary value problem on the half-line.

Paper B studies an initial-boundary value problem for the defocusing nonlinear Schrödinger equation on the half-line with asymptotically oscillatory boundary conditions. More precisely, the solution is required to approach a single exponential on the boundary as t → +∞ and to decay to zero as x → +∞. We construct a solution of the problem in a sector close to the boundary and compute its long-time behaviour.

Paper C studies a similar problem as Paper B but instead of the nonlinear Schrödinger equation we study the Gerdjikov-Ivanov equation. We give necessary conditions for the existence of a solution of the associated initial-boundary value problem under asymptotically oscillatory boundary conditions.

urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-294544

Page responsible:Kerstin Gustafsson
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Last changed: Jun 09, 2021