These notes provide an introduction to an approach to the theory of integrable systems that arises from the observation that many integrable systems are reductions of the anti-self-duality equations so that the theory of these equations can be understood as a reduction of the corresponding theory for the anti-self-duality equations.
We start with a general discussion of integrable systems and relations between them arising from symmetry reductions and give some standard examples. We then give an introduction to gauge theory and the self-dual Yang–Mills equations.
The anti-self-dual Yang–Mills equations can be seen to be an integrable system; it has a Lax pair, admits Backlund transformations, there are ansatze for solutions, and it has topological solutions, instantons. We go on to discuss its reductions to three dimensions, monopoles and Chiral models.
Another way to see the self-dual Yang–Mills equations as an integrable system is to present Hamiltonian and Lagrangian formulations and a recursion operator. This leads to the generalised self-dual Yang–Mills Hierarchies.
We then discuss general principals of reduction. Translation reductions lead to the KdV and nonlinear Schrodinger equations. Non translational reductions give rise to the Ernst equations and Painlevé equations. Finally we discuss further developments of the overview.
These notes are intended to provide an introduction to an overview on the theory of integrable systems based on reductions of the anti-self-dual Yang–Mills equations and its twistor construction.