| In the first part of this thesis we show that an approach similar to the one mentioned above for linear PDEs can also be used to solve initial-boundary-value problems (IBVPs) for a general class of discrete linear evolution equations (DLEEs). The method is quite general, and it works for many IBVPs for which Fourier or Laplace methods are not applicable. Even when such methods can be used, the present method has several advantages, in that it provides a representation of the solution which is convenient for both asymptotic analysis and numerical evaluation. Specifically, we present a transform method for solving IBVPs for linear semi-discrete (differential-difference) and fully discrete (difference-difference) evolution equations. The method is the discrete analogue of the one recently proposed by A.S. Fokas to solve IBVPs for evolution linear partial differential equations. We show that any discrete linear evolution equation can be written as the compatibility condition of a discrete Lax pair, namely, an overdetermined linear system of equations containing a spectral parameter. As in the continuum case, the method employs the simultaneous spectral analysis of both parts of the Lax pair, the symmetries of the evolution equation and a relation, called the global algebraic relation, that couples all known and unknown boundary values. The method applies for differential-difference equations in one lattice variable, as well as for multidimensional and fully discrete evolution equations. We demonstrate the method by discussing explicitly several examples.;In the second part of this thesis we build on the analytical tools that have been developed for the KP equation and present a much more general classification of the soliton solutions of the two-dimensional Toda lattice (2DTL) than was done in Ref. [130], pointing out both the similarities and the differences between the 2DTL and the KP equation. Specifically, we classify the soliton solutions of the 2DTL by exploiting the structure of the Casoratian expression for its tau function. In general, these solutions consist of unequal numbers of "incoming" and "outgoing" line solitons. We classify the incoming and outgoing line solitons based on asymptotic analysis of the tau function of the 2DTL as the discrete variable tends to infinity. We also identify various subclasses of solutions and characterize some of them in terms of the amplitudes and directions of the interacting solitons. As a special case, we obtain the reduction to the soliton solutions of the one-dimensional Toda lattice. Throughout, we point out the similarities with --- and the differences from --- the corresponding results for the KP equation.;The sine-Gordon (SG) is a ubiquitous physical model that describes nonlinear oscillations in a number of settings ranging from Josephson junctions, to self-induced transparency, to crystal dislocations, to Bloch wall motion of magnetic crystals and beyond (see [154] for references). The SG is also an important theoretical model as it is a completely integrable system [12], like the KP equation and the 2DTL. Many non-integrable nonlinear models also bear solitary wave solutions. In the specific case of the SG equation, these traveling wave solutions take the form of "kink". Because various physical effects often result in the presence of localized defects or impurities, the interaction of such solutions with defects has been extensively studied in the literature [53, 39, 55, 85, 87, 91, 92, 93, 94, 127, 136, 143, 146, 147, 157, 165].;Unlike the unperturbed SG, kink-plus-impurity system is not exactly solvable, and therefore many of its properties must be studied using numerical or asymptotic methods. In the last part of this thesis we investigate the properties of various numerical methods for the study of perturbed SG equation with impulsive forcing. In particular, we consider finite difference and pseudo-spectral methods for discretizing the SG equation, we discuss different methods of discretizing the Dirac delta function, and we use each combination of these methods to model the soliton-defect interaction. We present a comprehensive study of the convergence properties of all these different combinations of methods. We find that no single discretization method for the partial differential equation is the best in all cases. Instead, the properties of each method depend heavily on the specific representation chosen for the Dirac delta---and vice versa. (Abstract shortened by UMI.)... |
