Long-time asymptotics for the defocusing NLS

The project Long-time asymptotics for the defocusing nonlinear Schrödinger equation on the half- line with time-periodic boundary data aims to study the long-time behavior of solutions of the defocusing nonlinear Schrödinger equation (dNLS). This nonlinear partial differential equation serves as a universal model and arises in a vast number of applications, ranging from nonlinear fiber optics and nonlinear water waves to Bose-Einstein condensates. The question we are concerned about can be summarized as follows: what is the long-time effect of an external periodic drive on this nonlinear system? We explain this question with the help of an intuitive example. Let us imagine a rope hanging freely from a high building. We begin to shake the top end periodically, for instance according to a sine rhythm. How will the waves that we are generating at the top look like when they pass through the middle section of the rope, how will they behave shortly before they reach the end of the rope? What changes if we shake harder, or with a higher or lower frequency? Similar questions arise naturally in the context of waves governed by dNLS and generated by an external periodic driver. Answers to these questions are of fundamental theoretical interest, but also of practical importance, for instance when considering the transmission of optical signals through nonlinear optical fibers. Our goal is to find answers to this challenging problem, which has not been addressed in the research literature so far. This problem can be formulated mathematically in terms of a so-called initial boundary value problem: the periodic drive at one end is encoded with the help of a suitable boundary condition. To study the solutions of this problems, we will build upon a Riemann-Hilbert approach for initial boundary value problems for nonlinear integrable partial differential equations, and determine the long-time behavior of solutions of the related Riemann-Hilbert problem by using the powerful nonlinear steepest descent method.

This project was concerned with the defocusing nonlinear Schrödinger equation, which plays an important role in mathematics, mathematical physics, and in several additional fields in science and engineering. In its applications this equation describes the propagation of certain dispersive waves through nonlinear media; for instance, particular water or ocean waves, light waves in optical fibers or plasma waves. Solutions of the nonlinear Schrödinger equation describe the evolution and spacial properties of such systems, which underlines their importance. We focused specifically on solutions satisfying certain initial and boundary conditions. At its only boundary point, the problem under study is prescribed by an asymptotically periodic function of time, whereas solutions are supposed to decay to zero in the far (spatial) distance. One may think of a periodic input signal for a physical system, which is (almost) still in the distance. Since the underlying equation is a nonlinear partial differential equation, already the question about existence of such solutions is an extremely complicated mathematical problem. Moreover, we were interested in the long-time behavior of these solutions. Hence, the asymptotic properties of solutions were of special interest. With the help of spectral methods for so-called integrable nonlinear partial differential equations (like the nonlinear Schrödinger equation), we were able to construct solutions for the particular initial-boundary value problem under consideration. Moreover, we managed to establish the long-time behavior of solutions by deriving explicit approximation formulas. In particular we proved that there exist precisely three asymptotic sectors in the space-time plane, which give rise to different qualitative behavior of solutions: there is a plane wave sector close to the periodic "input signal", a sector of asymptotically decaying waves far away, and an area with damped modulated waves in between. Furthermore, we considered a related initial value problem. We studied the evolution of a specific initial wave, which asymptotes to a periodic form in one direction and decays to zero in the opposite direction. We proved comprehensive results for the corresponding spectral data, the global existence of solutions, and derived a detailed description of their long-time behavior.