Mapping Properties of Bäcklund Transformations and the Asymptotic Stability of Soliton Solutions for the Nonlinear Schrödinger and Modified Korteweg-de-Vries Equation

Abstract

We consider the cubic Nonlinear Schrödinger Equation (NLS) and the Modified Korteweg-de-Vries Equation (mKdV) in the one-dimensional, focusing case. For the mKdV, we also restrict ourselves to the case of real-valued solutions. The Lax system for the Nonlinear Schrödinger Hierarchy gives rise to a Bäcklund transformation, which connects the trivial zero solution to soliton solutions for both equations.<br /> Building upon work by Mizumachi and Pelinovsky, as well as asymptotic stability results for the zero solution by Ifrim and Tataru (NLS) and Harrop-Griffiths (mKdV), we prove asymptotic stability of solitons via the Bäcklund transformation. This provides an alternative to other approaches in the literature in that we do not invoke an explicit analysis of the Riemann-Hilbert problem for solutions close to zero. Even in the absence of the kind of "structural" information about solutions provided by asymptotic expressions or the inverse scattering formalism, the arguments developed in this thesis would (at least) yield a weakened, "preliminary" form of our results, which involves small, time-dependent position and phase shift functions. Much of the proof might be applicable to other equations in the focusing NLS hierarchy. We ultimately establish convergence properties of the Jost solutions for small NLS and mKdV potentials in the Lax system, leading to a more precise, quantitative understanding of stability properties.