Building Lax Integrable Variable-Coefficient Generalizations to Integrable PDEs and Exact Solutions to Nonlinear PDEs

This dissertation is composed of two parts. In Part I a technique based on extended Lax Pairs is first considered to derive variable-coefficient generalizations of various Lax-integrable NLPDE hierarchies recently introduced in the literature. It is demonstrated that the technique yields Lax- or S-integrable nonlinear partial differential equations (PDEs) with both time- and space-dependent coefficients which are thus more general than almost all cases considered earlier via other methods such as the Painleve Test, Bell Polynomials, and various similarity methods. However, this technique, although operationally effective, has the significant disadvantage that, for any integrable system with spatiotemporally varying coefficients, one must \u27guess\u27 a generalization of the structure of the known Lax Pair for the corresponding system with constant coefficients. Motivated by the somewhat arbitrary nature of the above procedure, we present a generalization to the well known Estabrook-Wahlquist prolongation technique which provides a systematic procedure for the derivation of the Lax representation. In order to obtain a nontrivial Lax representation we must impose differential constraints on the variable coefficients present in the nlpde. The resulting constraints determine a class of equations which represent generalizations to a previously known integrable constant coefficient nlpde. We demonstrate the effectiveness of this technique by deriving variable-coefficient generalizations to the nonlinear Schrodinger (NLS) equation, derivative NLS equation, PT-symmetric NLS, fifth-order KdV, and three equations in the MKdV hierarchy. In Part II of this dissertation, we introduce three types of singular manifold methods which have been successfully used in the literature to derive exact solutions to many nonlinear PDEs extending over a wide range of applications. The singular manifold methods considered are: truncated Painleve analysis, Invariant Painleve analysis, and a generalized Hirota expansion method. We then consider the KdV and KP-II equations as instructive examples before using each method to derive nontrivial solutions to a microstructure PDE and two generalized Pochhammer-Chree equations