Course title:   Introduction to Solitons     ( Karen  Uhlenbeck)

Time: Tu Th 2:00-3:15  Spring 2003

Web address: /home.html

Audience Prerequisites: First  and second year graduate students interested in geometry and/or applied mathematics

Prerequisites: Basic undergraduate material; especially differential equations and systems, matrix algebra and elementary mechanics

What is a soliton? Solitons are special solutions to equations which evolve in time with a fixed profile.  Many equations have solitons, but  multisolitons tend to occur in equations which are known as "integrable".  The work integrable refers to the fact that many of these equations  can be "explicitly solved via  simple methods such as integration". Integrable equations are very special, but they are heavily used in applied mathematics, geometry and mathematical physics, to say nothing of their rather unexpected appearances in the study of the topology of algebraic varieties (quantum cohomology), mathematical biology , combinatorics and so forth.

Course description:  This a course designed for first and second year graduate students who are interested in geometry (and/or applied mathematics). The goal is to show how various different theoretical and computational ideas are used in the study of phenomena which are physical, geometric and algebraic. We will discuss particular examples of partial and ordinary differential equations and their solutions, Lagrangian and Hamiltonian mechanics, conservation laws, scattering theory, matrix groups and algebras, and possibly a little of the geometry of curves as surfaces as well as an outline of why an applied mathematician might meet an elliptic curve or a theta function face to face.

Texts: Much  of the course will roughly follow the notes of Dick Palais .  I have also listed two other optional references.  The first duplicates the material in the notes to some extent.

Grades:   Students are expected to choose a topic and make a presentation on it, either individually or preferably in groups of two or three. (Did you ever notice how many math papers are written by two or three authors?) Problems will be offerred but not required.

Tentative Outline (includes modifications of the notes of Palais to include some basic examples from mechanics)

I. Model wave equations :
1) Linear waves, dispersion and dissipation and general solutions (or, why do we need Fourier series, Fourier transforms, Bessel functions and Airy functions?)
2) Solitons for non-linear second order equations (including Sine Gordon) (or why we need cosh, sinh and tanh).
3) Conservation laws and traffic pile-ups (shocks).

II. The Korteweg-de Vries equation :
1) History; What  did Russell, Korteweg-de Vries, Fermi-Pasta-Ulam and Zabusky- Kruskal do?.
2) Conservation laws
3) Hamiltonian formulation 4) The Lax pair

III. A digression into finite dimensional integrable systems (mechanics)
Conservation of energy
2) Conservation of momentum and angular momentum
3) Lagrangian and Hamiltonian mechanics
4) The two body problem; central force problems
5) Conservation laws due to general symmetries (incorrectly referred to as Noether's theorem)
6) Some brief remarks about symplectic manifolds and Hamiltonian flows

IV. Finite dimensional integrable systems
2) Model equations on matrix groups
3) The Toda lattice

IV. Back to KdV
1) The Hierarchy of flows for KdV
2) Multisoliton formulas
3) The scattering theory
4) The conversion to a 2x2 matrix theory
5) Remarks on the program for periodic solutions (enter algebraic geometry)

V. The Non-linear Schroedinger equation :
1) History (One of the major successes of mathematics in the 20th century is the use of NLS in fiber optics).
2) Evolution of a curve by the binormal; the Hashimota transform
3) Repeat of the program (depending on the time, energy and enthusiasm)

VI. The Sine-Gordon Equation (I doubt we will get this far!)
1) History, or surfaces in 3 space and 19th century geometry.
2) The matrix formulation and Lax pair
3) The Backlund transformations (loop group formulation)
4) A sketch of how chaos occurs in a  perturbation of a soliton family  (This is a famous series of papers by Ercolani, Forest and McLaughlin).