The defocusing nonlinear Schrödinger equation with step-like oscillatory data

Sammanfattning: 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.