Halis Yilmaz

Evolution equations for Differential Invariants.

(July 2003)

Supervisor: Chris Athorne

Abstract: This thesis is concerned with the invariant forms of nonlinear evolution equations and their Laplace transformations.

The thesis is organized as follows: Chapter one is an introduction to the concepts of Soliton Theory. We discuss the Inverse Scattering-Lax generalization and Hirota Bilinear Transformation as some solution methods to Evolution Equations such as Korteweg-de Vries (KdV), Kadomtsev-Petviashvili (KP), Davey-Stewartson (DS) and Novikov-Veselov (NV) equations. Here the KdV equation will be presented as an example to obtain solutions by using the above methods.

In chapter two we start with the definitions of invariant and covariant by following Hilbert's book [Hilbert, D. , Theory of algebraic invariants Cambridge Univ. Press, Cambridge, 1993] closely. We carry on by introducing gauge, Darboux and Laplace transformations respectively. Under the section Laplace transformations we calculate gauge invariants and their Laplace maps for a second order linear hyperbolic equations [Athorne, C. , Inverse Problems 9 (1993) 217--232; Konopelchenko, B.G, Phys. Lett.A, 156 (1991) 221--222; Zhiber, A.V. and Sokolov, V.V., Uspekhi Mat. Nauk 56 (2001), 63--106]. Then we give the definitions of gauge invariants and Laplace maps for an $n\times n$ linear differential operator matrix. We then move on the invariants and their Laplace transformations for the cases $n=2,3,4$, where the ${\mathbb Z}\times{\mathbb R}^2$, ${\mathbb Z}^2\times{\mathbb R}^3$ and ${\mathbb Z}^3\times{\mathbb R}^4$ Toda lattice equations respectively arise . Here the cases $n=2,3$ are obtained by Athorne [Athorne, C. , Phys. Lett.A, 206 (1995) 162-166.] In this chapter we also try to answer the question: {\em What is the relationship between the invariants of two $n \times n$ differential operator matrices which commute?} Here we consider the cases $n=2$ and $n=3$. We also discuss the completeness of invariants for $n\times n$ differential operator matrices, where we start with the cases $n=2,3$ and then generalize the method for the case $n \times n$. Finally we examine matrix covariants and their Laplace maps for a differential operator matrix ${\mathbb L}$, where we start with the case $m \times m$ [Konopelchenko; Sokolov ibid.] and then search the case $N\times N$ where $N=n+m$ and the entries of ${\mathbb L}$ are $n\times n$, $n\times m$, $m\times n$ and $m\times m$ matrices respectively.

In chapter three, we will calculate invariants for scalar operators by using the gauge transformation. Here the Lax equation and `$L-M-f$' triad representation lead us to determine the invariant forms of the related equations. We present the KP and NV equations as the examples of the invariant form for scalar evolution equations. We will also discuss the completeness of the set of invariants for the operators $L$ and $M$.

In chapter four, we will calculate invariants for linear $2\times 2$ matrix operators of orders 2 and 3 (AKNS Lax operators) and give results on the associated evolution equations in invariant form like the DS and NV equations. We shall also discuss the compatibility of these flows with Laplace transformations. At the end of this chapter a solution of the NV equation will be given as an example to construct solutions to more general equations by using the Laplace maps. In chapter five we present results on the completeness of the set of invariants for a general hyperbolic system. It is simplest to do this by working in the off-diagonal gauge.

Chapter six is the critical summary of the thesis and contains comments on the results obtained throughout. In this chapter we also pose some open questions.

Publications:

1. The geometrically invariant form of evolution equations (with C.Athorne), J. Phys. A: Math. Gen. 35 (2002) 2619--2625