Ma 191c-sec7:  Integrable Systems (Spring 2015-16)

ANNOUNCEMENTS


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.


PREREQUISITES

There are no prerequisites.


SCHEDULE

TR 14:30 - 15:55, 257 Sloan.


INSTRUCTORS

Christopher Ormerod
TAUSSKY-TODD INSTRUCTOR IN MATHEMATICS
284 Sloan
626-395-4831


TA's

TBA


OFFICE HOURS

Wednesday 3:00pm


POLICIES

TBA


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.


READING