Boundary value problems for linear and nonlinear wave equations

It is well known that the Fourier transform can be used to solve initial value problems (IVPs) for linear evolution equations (LEEs). It is also well known that Fourier sine or cosine transforms can be used to solve certain initial-boundary value problems (IBVPs) for some LEEs. Unfortunately, not all IBVPs can be solved in this way, even for linear partial differential equations (PDEs). For example, while the IBVP for the linear Schrodinger equation (LS) on the half line can be solved via Fourier sine and cosine transforms, for the linear Korteweg-deVries equation on the half line it turns out that there are no proper analogues of sine and cosine transforms that allow the solution of the IBVP on the half line, even though the problem is perfectly well posed. Moreover, Fourier transform methods are unable to solve IVPs for nonlinear evolution equations (NLEEs). However, there exists certain NLEEs, called integrable systems, for which a nonlinear analogue of the Fourier transform, called the inverse scattering transform (IST), exists. The most well-known examples of such kind of systems are the nonlinear Schrodinger equation (NLS), the Korteweg-deVries equation (KdV), and the sine-Gordon equation (sG). It should be also noted that these equations are important not only because they display a surprisingly rich and beautiful mathematical structure, but also because they frequently arise as fundamental models in a variety of physical settings.;Following the solution of IVPs via IST, a natural issue was the solution of IBVPs for integrable NLEEs. After some early results, however, the issue remained essentially open for over twenty years. Recently, renewed interest in solving IBVPs for integrable evolution equations has lead to a number of developments. Particularly important among these is the method developed by A. S. Fokas, sometimes called the Fokas method, which is a significant extension of the IST. The method takes advantage of the Lax pair formulation, and is based on the simultaneous spectral analysis of both parts of the Lax pair, constructing both x and t transform. A crucial role is also played by a relation called global algebraic relation that couples all known and unknown boundary values. Indeed, it is the global relation that makes it possible to eliminate the unknown boundary data that appear in the integral representation of the solution for the IBVP.;The main purpose of this thesis is to solve IBVPs for continuous and discrete evolution equations. First, we solve discrete linear and integrable discrete NLEEs via the spectral method proposed by A. S. Fokas. To this end, we demonstrate the method to solve discrete LS equation and the IDNLS equation on the natural numbers, which can be referred to as the discrete analogue of IBVPs for PDEs on the half line. Even though additional technical complications arise in discrete problems compared to continuum ones, we show that a similar approach can also solve IBVPs for linear and integrable nonlinear DDEs. In the linear case we also explicitly discuss Robin-type BCs not solvable by Fourier series. In the nonlinear case we also identify the linearizable BCs and we discuss the elimination of the unknown boundary datum.;Next, we characterize the soliton solutions of the NLS equation via the nonlinear method of images. Using an extension of the solution to the whole line and the corresponding symmetries of the scattering data, we identify the properties of the discrete spectrum of the scattering problem. We show that discrete eigenvalues appear in quartets as opposed to pairs in the IVP, and we obtain explicit relations for the norming constants associated to symmetric eigenvalues. This means that for each soliton in the physical domain, there a symmetric counterpart exists with equal amplitude and opposite velocity. As a consequence, solitons experience reflection at the boundary. The apparent reflection of each soliton at the boundary of the spatial domain is due to the presence of a "mirror" soliton, located beyond the boundary. We then calculate the position shift of the physical solitons as a result of the nonlinear reflection. These results provide a nonlinear analogue of the method of images that is used to solve boundary value problems in electrostatics. (Abstract shortened by UMI.)...