Mark Ratter

Grammians in nonlinear evolution equations

July, 1998

Supervisor: Claire Gilson

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 interaction equation. More precisely we are interested in the role that Gramians play in determining new solutions to three-dimensional three-wave resonant interactions (3D3WR), through Hirota's bilinear method [Hirota, R., Direct methods in soliton theory, (1980) 123-146, Springer-Verlag] and the binary Darboux transformation [Matveev, V.B. and Salle, M.A., Darboux transformations and solitons, (1991) Springer-Verlag; MathSciNet Review]. 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 to provide an overall framework for the work that follows and an outline of the connections between the chapters. We also try to consider the motivation for working with the Grammian approach.

In chapter 2 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 [Manakov, S.V. et al., Phys. Lett. A, 63 (1977) 205-6][Ablowitz, M.J. and Satsuma, J., J. Math. Phys. 19 (1978) 2180-6; MathSciNet Review], and Gilson and Nimmo's work [Proc. Roy. Soc. Lon. A 435 (1991) 339-357; MathSciNet Review] 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 realxing the necessary reality conditions we may obtain rational solutions from ageneral 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 [Satsuma, J. and Ablowitz, M.J., J. Math. Phys. 20 (1979) 1496-1503; MathSciNet Review]. 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 [Physica D1 (1980) 45-67; MathSciNet Review] [J. Math. Phys. 22 (1981) 1176-81; MathSciNet Review] can easily be recast in terms of Grammians. This approach arises naturally by considering the Painlevé analysis for 3D3WR [Ganesan, S. and Lakshmanan, M.,J. Phys. A 20 (1987) L1143-7; MathSciNet Review] through which we recover Kaup's Bäcklund transformations and the bilinear form. Kaup's solutions are generalized to give the n-lump solution and then we prove a general Grammian solution by using a Jacobi dentity. Finally in chapter three we examine some specific examples of the lump solutions and provide some idae of what the solutions look like. The work in this chapter constitutes reference [1] below.

We stay with 3D3WR in chapter four. By focussing on its scattering problem and using the method developed by Nimmo we derive Darboux transformations (DT) and binary Darboux Transformations (BDT). It turns out that only the BDT preserves 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 Bäcklund transformation of Kaup [J. Math. Phys. 22 (1981) 1176-81; MathSciNet Review] 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.

Publications:

[1] Gilson, C.R. and Ratter, M.C., J. Phys. A 31 (1998) 349-67; MathSciNet Review

©MCRatter,1998