## ANNOUNCEMENTS

- Beginning of Instruction for the Spring term is Monday, March 28, 2016.
- This page was last updated: .

## COURSE DESCRIPTION

This a course on integrable systems. We start with a treatment of integrable PDE's, with a particular focus on wave equations, most importantly, the KdV equation. This includes a discussion of soliton solutions, symmetries, conserved quantities and various methods for solving the KdV equation such as direct linearization, inverse scattering and Hirota's bilinear method. We proceed to a treatment of ODE's, mainly on the Painleve equations, topics such as isomonodromy, symmetries, special solutions and applications. Depending on time and the interests of the group, we shall delve into discrete integrable systems.

## TOPICS COVERED

The history of Solitons.

Travelling wave solutions.

Symmetries of
the KdV equation.

Lax Integrability.

Conserved quantities.

Inverse
Scattering.

Hirota's Bilinear method.

Direct Linearization.

Painleve tests

Backlund transformations and special solutions

Isomonodromy

## TEXTBOOKS

I am working from a number of books.

Peter Olver: Applications of Lie Groups to Differential equations

Mark Ablowitz and Peter Clarkson: Solitons, Nonlinear Evolution Equations
and Inverse Scattering

Jimbo, Miwa and Date: Solitons: differential
equations, symmtries and infinite dimensional algebras.

## LECTURE NOTES

The following lecture notes have somewhat hastely been put together. I wish to expand upon some section, however, at this point, I would welcome any additions, corrections and suggestions.

Date | Description |
---|---|

Notes 1 : KdV - travelling wave solutions. | |

Notes 2 : KdV - Symmetries of the KdV equation | |

Notes 3 : KdV - Group invariant solutions and Lax pairs | |

Notes 4 : KdV - Conservation laws and Backlund transformations | |

Notes 5 : KdV - Hirota's Bilinear method and direct linearization | |

Notes 6 : KdV - Inverse Scattering | |

Notes 7 : Painleve - The Painleve property and Painleve equations | |

Notes 8 : Painleve - Hamiltonian and Symmetries, PII and PIV | |

Notes 9 : Painleve - Isomonodromy and Schlesinger equations | |

Notes 10 : Painleve - The sixth Painleve equation | |

## Grading

Given the small class size, I intend to make the grading based on two student presentations on integrable systems not covered in the course.