Analytic Solutions to the NLS, MKDV, and KP Equations

Abstract

This dissertation comes in 3 parts. The first consists of a detailed study of the 1 + 1 MKDV withNZBC. A derivation of MKDV from its Lax Pair is given. Then we construct the Hamiltonian of defocusing MKDV from which we obtain regular, singular, and kink solutions. I apply the Miura Transformation MKDV to KDV via the Miura Transformation to obtain. Lastly, I give a derivation of regular and singular 1-soliton and 2-soliton solutions in the focusing case using the Dressing Method and extract the additive phase shift from long time asymptotic behaviour. The second part presents a framework for obtaining analytic solutions of the KP-1 equation using τ -functions. Solutions are obtained by taking the determinant of a Gram matrix of generalized Schur polynomials. A classification system of KP-1 solitons is implemented in terms of the degree of the generalized Schur polynomial, known as the depth, and the spectral parameter. Known results are recreated and two forms of nonlinear interactions are explored. The last part presents a new class of semi-rational solutions to the Davey-Stewartson equations obtained using yet another form of the Dressing Method. I express the kernel of the Marchenko equation in terms of factors of the spectral function obtaining ratios of Hermite functions, which after applying reductions, resulting in spatially-periodic solutions.