# Geometric structures on the target space of Hamiltonian evolution equations

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## Abstract

This thesis is concerned with the relationship between integrable Hamiltonian partial differential equations and geometric structures on the manifold in which the dependent variables take their values. Chapters 1 and 2 are introductory chapters, and as such contain no original material. Chapter 1 covers some basic material from the theory of integrable systems, including the Hamiltonian formalism for PDE's, the concept of a bi-Hamiltonian system, and the dispersionless Lax equation. Chapter 2 is about Frobenius manifolds. It explains their relationship to the WDVV equations of topological quantum field theory, and how they form part of the theory of integrable systems via both the deformed Levi-Civita connection and a flat pencil of metrics. Chapter 3 is based on [J. T. Ferguson. Flat pencils of symplectic connections and Hamiltonian operators of degree 2. J. Geom. Phys., 58(4):468–486, 2008]. It is original, except for the background material in Section 3.1. In it we explain the (almost) symplectic geometry associated to Hamiltonian operators of degree 2, and use it to formulate the geometric conditions for two such operators to constitute a bi-Hamiltonian structure. In the case that these operators are associated to symplectic forms, these conditions are expressed as algebraic constraints on a multiplication of one-forms. We also express conditions for a Hamiltonian operator of degree two to be compatible with a hydrodynamic type Hamiltonian operator. Chapter 4 is based upon [J. T. Ferguson and I. A. B. Strachan. Logarithmic deformations of the rational superpotential/ Landau-Ginzburg construction of solutions of the WDVV equations. arXiv:math-ph/0605078, 2006], which was joint work with Ian Strachan. It is to appear in Communications in Mathematical Physics, and is original except for Section 4.1. It is concerned with the construction of new solutions to the WDVV equations which arise by analogy with the so-called waterbag reductions of the dispersionless KP hierarchy. Superpotentials of existing Frobenius manifolds are deformed by the addition of logarithmic terms, and this results in new WDVV solutions which deform existing ones, including a new class of polynomial solutions which deform solutions associated to the A_N Coxeter group. Chapter 5 follows on from Chapter 4, and considers in detail two integrable hierarchies which arise from the WDVV solutions studied there. It is particularly concerned with the Hamiltonian structures of these hierarchies. Appendix A attempts to incorporate some of the features of one of these hierarchies into a construction of a Frobenius structure from a bi-Hamiltonian structure.

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