Boundary value problems for discrete and continuous nonlinear Schrödinger equations
It is well known that the Fourier transform can be used to solve initial value problems (IVPs) for linear partial differential equations (PDEs). It is also well known that a special class of nonlinear PDEs exists for which a nonlinear analogue of this technique exists, called the inverse scattering transform (IST). Equations of this type, usually called integrable systems, exhibit a surprisingly rich and beautiful mathematical structure. A large body of knowledge has been accumulated on these systems over the last forty years. In particular, the solution of IVPs for integrable nonlinear PDEs in one spatial and one temporal dimension was developed in the 1970's under the assumption of rapidly decaying initial conditions at infinity, hereafter referred to as zero boundary conditions (ZBCs). On the other hand, IVPs in which the initial condition satisfies non-zero boundary conditions (NZBCs) at infinity are much less well characterized. So are boundary value problems (BVPs). In particular, BVPs for integrable nonlinear PDEs can only be linearized for a special kind of boundary conditions (BCs); such BCs are then called linearizable. Both of these kinds of problems (namely, IVPs with NZBCs and BVPs) are still the object of active research. This thesis is devoted to both kinds of problems. Specifically, the thesis contains the following original contributions: I. In Chapter 3 we revisit various problems for the focusing nonlinear Schrödinger (NLS) equation with ZBCs at infinity. Explicitly, we present a detailed discussion of: (i) double poles in the scattering problem within the framework of the Riemann-Hilbert formalism; (ii) BVPs on the half line with linearizable BCs at the origin, including self-symmetric eigenvalues and the reflection-induced position shift. II. In Chapter 4 we develop a method to solve BVPs for the Ablowitz-Ladik system on the natural numbers with linearizable BCs at the origin. We do so by constructing a suitable Bäcklund transformation. Importantly, this method allows us to deal efficiently with self-symmetric eigenvalues. As a result, we completely classify the solutions to the BVP of the AL system with a linearizable BC at the origin. III. In Chapter 5 we develop a method to solve BVPs for the defocusing NLS equation on the half line with NZBCs at infinity and linearizable BCs at the origin. As with the Ablowitz-Ladik system, we do so by constructing a suitable Bäcklund transformation, and we use it to completely characterize the BVP, including the self-symmetric eigenvalues. IV. In Chapter 6 we provide a detailed comparison of two different approaches to the IST for the defocusing vector NLS (VNLS) equation with NZBCs at infinity. After briefly reviewing the standard IST approach developed in  for the two-component VNLS equation and the new approach to IST used in  for the multi-component VNLS equation, we show how the new approach relates to the standard one for both the scalar NLS equation and the two-component VNLS equation. These results serve both to obtain a better understanding on the new approach, and as a preparatory step to obtain explicit soliton solutions in the multi-component case. 65, 66. Please see dissertation for references.