Kevin Baron (UK)

**Thesis title: **"Deformations of equations of hydrodynamic type "

**Date: **2006

**Supervisor:** Ian Strachan

**Abstract:** The main aim of this thesis is two-fold. Firstly it
shows how one utilises the so-called Modified Variables to analyze
many interesting integrable systems of hydrodynamic type. More
specifically we study the dispersionless Toda (dToda) system and
the Gas Dynamics equations from this viewpoint.

We also study deformations of multi-component equations of hydrodynamic type, and show how one can classify such deformations through certain obstructions that arise through a commuting flow-type technique.

Chapter One is a basic introduction to Soliton theory and encounters a broad range of topics in this area. Hamiltonian and bi-Hamiltonian structures are presented for the first time here, a notion of bi-Hamiltonian structures is a necessity for study of deformations in Chapter Four.

Chapter Two considers dispersionless Integrable systems and moves onto the deformations of such systems. The set-up needed for deformations in Chapter Four is given here, all of the technical details, as well as examples, are also given.

Chapter three introduces the reader to the role of the
modified variables and shows how one utilises them to be able to
construct Lax hierarchies of so-called non-standard type. Here we
show the equivalence of the dToda hierarchy (as a reduction of the

Benney hierarchy) in standard and non-standard form. The modified
variables allows one to reduce this problem to a combinatoric
argument, cutting down many computations.

Chapter Four discusses the deformation of
multi-component equations of hydrodynamic type. Here we use the
machinery defined in Chapter Two to deform a multi-component

KdV-type equations and related the higher order deformations to
Poisson cohomology groups. A detailed account of this will be
given and is very laborious in parts. This was considered by the
author as a necessary part of the Chapter in order the reader

fully appreciates the calculations involved. This Chapter
considers first order and second order deformations, showing how
one classifies the deformations in terms of arbitrary

functions.

Chapter Five considers the deformation of the dToda
system and the Gas dynamics equations from the point of view
of keeping the integrability via commuting flows. The

obstructions to the deformations will be shown to be classified in
terms of an unknown function, this function shall be given in
terms of known functions in the Hamiltonian generating the zeroth
order system (i.e; the normal evolution equations).

Finally, Chapter Six will summarize the key results of the thesis and gives ideas for possible further research problems.

Halis Yilmaz (Turkey)

**Thesis title: **"Evolution equations for Differential Invariants"

**Date: **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:**

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

William B. Dickson (USA)

**Thesis title:** "Branched 1-Manifolds and Presentations of Solenoids"

**Date:** 1998

**Supervisor:** Chris Athorne

**Abstract: **The aim of the project was to study the presentations of solenoids by branched 1-manifolds. We begin by studying two properties of branched 1-manifolds which effect the presentation of solenoids, orientability and recurrence. Solenoids are shown to come in two varieties, those presented by orientable branched 1-manifolds and those presented by non-orientable branched 1-manifolds. Two methods of determining whether or not a branched 1-manifold is orientable are given. Recurrence is shown to be a necessary and sufficient condition for a branched 1-manifold to present a solenoid. We then show how the question of whether or not a branched 1-manifold is recurrent can be converted into a question in graph theory for which there exist efficient algorithms.

Next we consider two special types of presentation, elementary presentation and (p,q)-block presentation, which allow us to extract algebraic invariants for the equivalence of solenoids. A slightly stronger version of a result of Williams [*Topology* **6** (1967) 473-487; MathSciNet Review] is obtained which states that any solenoid with a fixed point is equivalent to one presented by an elementary presentation. The proof is constructive and gives a method for finding an elementary presentation given a presentation of a solenoid with a fixed point. Williams has shown [ibid.] that there is a complete invariant for the equivalence of solenoids given by an elementary presentation in terms of the shift equivalence of endomorphisms of a free group. We prove a result which shows that every solenoid can be given by a coutably infinite number of equivalent (p,q)-block presentations. The (p,q)-block presentations again allow us to find invarients for the equivalence of solenoids in terms of the shift equivalence of endomorphisms of a free group.

Finally we consider further invariants for the equivalence of solenoids which are derived from the endomorphism invariants. First we examine the invariants which arise upon abelianizing the free group in question. Second we introduce new invariants which reflect some of the non-abelian character of these endomorphisms. These new non-abelian invariants are then used to solve a problem posed by Williams [ibid.]

© WBDickson, 1998.

Thomas Hartl (Germany)

**Thesis title:** "Symmetry analysis of differential equations from a geometric point of view"

**Date:** 1998

**Supervisor:** Chris Athorne

**Abstract: **The central theme of this work is the step by step integration of Frobenius and Darboux type differential problems by quadratures. Building on the concept of a solvable structure of vector fields introduced by Basarab-Horwath [*Ukr. Jour. Math.* **43** (1991) 1330-7; MathSciNet Review], we demonstrate the wider role played by chains of relatively closed ideals of vector fields or one-forms (under the Lie bracket or the exterior derivative respectively) in this context. We extend the solvable structure concept to one-forms and develop the notion of solvable structures compatible with a Poisson structure. In the preocess we shed light of certain types of so-called hidden symmetries and some other known types of symmetry reduction. In the introductory sections we also introduce a new way of defining the exterior derivative, slightly generalise the theorem of Frobenius and present a concise way of characterising Poisson tensors by giving an explicit expression for the auto-Schouten bracket of a two-vector. Also included is a classification result drawing attention to the wide scope of non-Lie point symmetries.

©THartl,1998.

**Publications: **

- "
*Solvable structures and hidden symmetries*", (with C.Athorne), J. Phys. A 27 (1994) 3463-3474 -MathSciNet Review.

Mark Ratter (UK)

**Thesis title:** "Grammians in nonlinear evolution equations"

**Date:** 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.

©MCRatter,1998

**Publications: **

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

Metin Ünal (Turkey)

**Thesis title:** "Applications of Pfaffians to Soliton theory"

**Date: **1998

**Supervisor:** Jon Nimmo

**Abstract: **This thesis is concerned with solutions to nonlinear evolution equations. In particular we examine two soliton equations: the Novikov-Veselov-Nithzik (NVN) equations and the modified Novikov-Veselov-Nithzik (mNVN) equations. We are interested in the role that determinants play in determining new solutions to various soliton equations. The thesis is organised as follows.

In Chapter 1 we give an introduction and historical background to the soliton theory and recall John Scott Russell's observation of a solitary wave in 1844. We explain the Lax method and Hirota method and discuss the relevant basic topics of soliton theory that are used throughout this thesis. We also discuss different types of solutions that are applicable to nonlinear evolution equations in soliton theory. These are Wronskians, Gammians and Pfaffians.

In Chapter 2 we give an introduction to Pfaffians which are the main elements of this thesis. We give the definition of a Pfaffian and a classical notation for the Pfaffians is also introduced. We discuss the identities of Pfaffians which correspond to the Jacobi identity of determinants. We also discuss the differentiation of Pfaffians whcih is useful in the Pfaffian technique. By applying the Pfaffian technique to the BKP equation an example of soliton solutions to the BKP equation is also given.

In Chapter 3 we study the asymptotic properties of dromion solutions written in terms of Pfaffians. We apply the technique that is used in [Gilson, C. and Nimmo, JJC., Proc. Roy. Soc. Lon. A 435 (1991) 339-357; MathSciNet Review] for the Davey-Stewartson (DS) equtaions to the NVN equations. We study the asymptotic properties of the (1,1)-dromion solution and generalize them to the (2,2)-dromion solution and to the (2,1)-dromion solution and show the asymptotic calculations explicitly for each dromion. In the last section we give a number of plots which show various kinds of dromion scattering. These illustrate that dromion interaction properties are different from the usual soliton interactions.

In Chapter 4 we exploit the algebraic structure of the soliton equations and find solutions in terms of fermion particles [Jimbo, M. and Miwa, T. RIMS, Kyoto Univ. 19 (1983) 943-1001; MathSciNet Review]. We show how determinants and Pfaffians arise naturally in the fermionic approach to soliton equations. We write the tau function for charged and neutral free fermions in terms of determinants and Pfaffians respectively and show that these two concepts are analogous to one another. Examples of how to get soliton and dromion solurions from tau functions for the various soliton equations are given.

In Chapter 5 we use some results from [Nimmo, JJC., in Applications of Analytic and Geometric Methods to Nonlinear Differential Equations, Ed. P.A.Clarkson, pp 183-92 (1993), Kluwer; MathSciNet Review] and [Nimmo, JJC., Phys. Lett. A 168 (1992) 113-9; MathSciNet Review]. We study two nonlinear evolution equations: the Konopelchenko-Rogers (KR) equations and the modified Novikov-Veselov-Nithzik (mNVN) equations. We derive a new Lax pair for the mNVN equations which is gauge equivalent to a pair of operators. We apply the Pfaffian technique to KR and mNVN equations and show that these equations in the bilinear form reduce to a Pfaffian identity.

In this thesis, Chapter 1 is a general introduction to soliton theory and Chapter 2 is an introduction to the main element of this thesis. The contents of these chapters are taken from various references as indicated throughout the chapters. Chapters 3, 4 and 5 are the author's own work with some results used from other references also indicated in the chapters.

©MUnal, 1998