**Time: **Tu Th 2:00-3:15 Spring
2003

**Web address: **
www.ma.utexas.edu/users/uhlen/solitons /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.

- You can download "Introduction to Wave Equations and Solitons" as a pdf file by clicking. You can also download them directly from www.ma.utexas.edu/users/uhlen/solitons/notes.pdf or from the link from my home page. The course will introduce or review (depending on background!) basic concepts such as Fourier Transform, kinetic and potential energy, hamiltonian and lagrangian mechanics etc. although the notes tend to assume this background material.
- P.G. Drazin and R.S. Johnson, "Solitons: an introduction", Cambridge Texts in Applied Mathematics, Cambridge University Press, (l993). This is optional. It is paperback, but still probably outrageously expensive.
- A book on mechanics. This is optional, but highly recommended. Watch the home page for a list of suggestions.

**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

1)Definition

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).