# Part II - Integrable Systems

## Lectured by A. Ashton, Michaelmas 2016

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# Contents

- V Full version
- 0 Introduction
- 1 Integrability of ODE's
- 1.1 Vector fields and flow maps
- 1.2 Hamiltonian dynamics
- 1.3 Canonical transformations
- 1.4 The Arnold-Liouville theorem
- 2 Partial Differential Equations
- 3 Inverse scattering transform
- 3.1 Forward scattering problem
- 3.2 Inverse scattering problem
- 3.3 Lax pairs
- 3.4 Evolution of scattering data
- 3.5 Reflectionless potentials
- 3.6 Infinitely many first integrals
- 4 Structure of integrable PDEs
- 4.1 Infinite dimensional Hamiltonian system
- 4.2 Bihamiltonian systems
- 4.3 Zero curvature representation
- 4.4 From Lax pairs to zero curvature
- 5 Symmetry methods in PDEs