Grammians in Nonlinear Evolution Equations

Ratter, Mark C (1998) Grammians in Nonlinear Evolution Equations. PhD thesis, University of Glasgow.

Abstract

This thesis is concerned with solutions to nonlinear evolution equations. In particular we examine two specific equations; the Davey-Stewartson (DS) equation and the three-dimensional three-wave resonant interaction equation. More precisely we are interested in the role that grammians play in determining new solutions to three-dimensional three-wave resonant interactions (3D3WR), through Hirota's bilinear method [42] and the binary Darboux transformation [70]. We also exploit the grammian structure to obtain rational solutions to the DS equation. The thesis is organised as follows. Chapter one is an introduction to the concepts, ideas and constructions that will be used throughout this thesis. We discuss bilinear equations, Laplace expansions of determinants and grammians, all with a view to their role in obtaining solutions to nonlinear evolution equations. The chapter attempts to provide an overall framework for the work that follows and an outline of the connections between the chapters. We also try and consider the motivation for working with a grammian approach. In chapter two we focus on the DS equation with non-zero background, and in particular rational solutions for it. After background material to the DS equation and its derivation, we look more closely at methods that already exist to obtain solutions. Our aim is to provide a simple way to calculate rational solutions to the DS equation. The example of the KP equation [66] [7], and Gilson and Nimmo's work [35] provides the approach we need. We verify a broad class of solutions all written in terms of a grammian and from these we obtain singular rational solutions by exploiting the "long-wave limit". However, by relaxing the necessary reality conditions we may obtain rational solutions from a general grammian. By then verifying when these are solutions to the DS equation we obtain a wider class of rational solutions. This mirrors the approach of Ablowitz and Satsuma [89]. It leads us to determine a class of non-singular rational solutions which describe multiple collisions of lumps. These lumps correspond to the ones found by Ablowitz and Satsuma but the grammian method is simpler and the solutions more "fully" rational. In chapter three we consider 3D3WR using a bilinear approach to investigate a broad class of solutions. The solutions to 3D3WR described originally by Kaup [52] [51], can easily be recast in terms of grammians. This approach arises naturally by considering the Painleve analysis for 3D3WR [31], through which we recover Kaup's Backlund transformation and the bilinear form. Kaup's solutions are generalised to give the n-lump solution, and then we prove a general grammian solution by using a Jacobi identity. Finally in chapter three we examine some specific examples of the lump solutions and provide some idea of what the solutions look like. The work in this chapter constitutes [37]. We stay with 3D3WR in chapter four. By focusing on its scattering problem and using the method developed by Nimmo [78] we derive Darboux transformations (DT) and binary Darboux transformations (BDT). It turns out that only the BDT pre-serves the structure that we need for a solution to 3D3WR and these are written in a grammian format. By determining a closed form of the solution to the iterated BDT we see that it corresponds to the lump solutions of chapter three. This provides a link between the Backlund transformation of Kaup [51] and the BDT. We look briefly at obtaining a discrete version of 3D3WR from the BDT. Chapter five seeks to bring together the results of the various chapters and again identify the common theme of the grammian. We also discuss some open questions that arise from the work presented.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Additional Information: Adviser: Claire Gilson
• Keywords: Mathematics
• Date of Award: 1998
• Depositing User: Enlighten Team
• Unique ID: glathesis:1998-76448
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 19 Nov 2019 14:20
• Last Modified: 19 Nov 2019 14:20
• URI: https://theses.gla.ac.uk/id/eprint/76448