1 - Introduction

Summary

Background

Integrable systems are systems of partial or ordinary differential equations that combine nontrivial nonlinearity with unexpected tractability. Often one can find large families of exact solutions, and general methods for generic solutions. This volume is concerned with the deep links that integrability has with geometry. There are two rather different ways that geometry emerges in the study of integrable systems.

Geometrical context for integrable equations

The first is from the context of the differential equations themselves: even those integrable equations whose origins, perhaps in the theory of water waves or plasma physics, seem a long way from geometry can usually be expressed in the context of symplectic geometry as possibly infinite dimensional Hamiltonian systems with many conserved quantities and often with much more further structure. But geometry is itself also a rich source of integrable systems; one of the first examples of a completely integrable nonlinear partial differential equation, the sine-Gordon equation first appeared in the 19th century theory of surfaces, as a formulation of the constant mean curvature condition on a 2-surface embedded in Euclidean 3-space. Now there are many more examples from geometry in many dimensions, from the two-dimensional systems given by harmonic maps from Riemann surfaces to symmetric spaces, to the anti-self-duality equations in 4-dimensions and more generally quaternionic structures in 4k-dimensions.

The contributions of Tod, Mason and Woodhouse focus on the anti-self-duality equations either on a Yang–Mills connection on a vector bundle over ℝ⁴, or on a 4-dimensional conformal structure.