Integrable initial-boundary value problems and solution methods for the nonlinear Schrödinger equation

In this thesis, we review the inverse scattering transform with zero and non-zero boundary conditions at infinity for the one-dimensional nonlinear Schrödinger equation. The inverse problems are discussed making use of the theory of Riemann-Hilbert problems. We perform the analysis of the focusing nonlinear Schrödinger equation on the half-line with time-dependent boundary conditions at the origin and zero boundary conditions at infinity along the lines of the nonlinear mirror image method with the help of Bäcklund transformations. We find two possible classes of solutions. One class is very similar to the case of Robin boundary conditions whereby solitons are reflected at the boundary, as a result of effective interaction with their images on the other half-line. The new class of solutions supports the existence of one soliton that is not reflected at the boundary but can be either absorbed or emitted by it. We demonstrate that this is a unique feature of time-dependent integrable boundary conditions. Finally, we present partial results of the analysis for the focusing nonlinear Schrödinger equation on the half-line with Robin boundary conditions at the origin and non-zero boundary conditions at infinity using the nonlinear mirror image method in conjunction with Bäcklund transformations