What is Soliton Theory?

$KdV 2-Soliton Collision$

Professor: Alex Kasman
kasmana@cofc.edu

KdV Revolution In the 1960's, we finally realized that there was something interesting and unusual about the KdV equation. It had these "soliton" solutions that look like interacting bumps...almost like particles even though they are really waves. It was solvable in that we could write down its solutions exactly even though it was nonlinear. Also, even though it was nonlinear, it was almost as if the different 1-soliton solutions could be combined to make new solutions...like a nonlinear version of the linear superposition rule.

Answer to the Question: This semester, the title of our capstone class is Math 495: Soliton Theory. So, it is reasonable to as "what is that?"
Soliton theory is the subject that grew out of these observations. It seeks to answer the questions "What is special about KdV and are there other equations like it?" As it turns out, there are lots of ways to understand the special structure of the KdV equation, and there are lots of other equations with the same properties. As we discover them, study them, solve them and find applications for them in the real world, we call all of these results "soliton theory".

Structure of KdV The incredibly rich structure of the KdV equation can be understood from many different perspectives, like the many facets of a beautiful gem. In some way, they reflect some common property, but they are really independent and can at least sometimes be found in isolation. Understanding their relationship to each other is one goal of soliton theory that has already helped us to better understand mathematics itself. To get an idea, look at the "methods of solution" below.

Methods to "Solve" KdV Here is a (partial) list of methods to produce solutions to the KdV equation. I am certain that some experts would disagree with my choices, since I may have left off some important methods (e.g. "prolongation") and also have listed separately some closely related ideas (e.g. Darboux vs. Backlund Transformations). But, the point is just to illustrate the diversity of methods available.
I have put a "(*)" at the end of those that I am hoping to be able to introduce in this class.

Analysis

Inverse Scattering
Riemann-Hilbert problems
Painleve Property

Algebra

Lax pairs (*)
Rank one conditions (*)
Differential Algebra / Darboux Transformation (*)
Symmetry Methods (*)
Backlund Transformations
Loop Group Factorizations

Algebraic Geometry

Grassmannian Manifolds / Plücker Relations (*)
Algebraic Curves (*)
Burchnall-Chaundy Theory of Commuting Differential Operators / finite gap potentials
Surfaces of constant curvature

Direct / Hirota Bilinear Method (tau-functions) (*)
Hamiltonian Systems

Conservation Laws ("integrals of motion") (*)
Quadrature
Symplectic Geometry / bi-Hamiltonian Structure

Professor:	Alex Kasman kasmana@cofc.edu
KdV Revolution	In the 1960's, we finally realized that there was something interesting and unusual about the KdV equation. It had these "soliton" solutions that look like interacting bumps...almost like particles even though they are really waves. It was solvable in that we could write down its solutions exactly even though it was nonlinear. Also, even though it was nonlinear, it was almost as if the different 1-soliton solutions could be combined to make new solutions...like a nonlinear version of the linear superposition rule.
Answer to the Question:	This semester, the title of our capstone class is Math 495: Soliton Theory. So, it is reasonable to as "what is that?" Soliton theory is the subject that grew out of these observations. It seeks to answer the questions "What is special about KdV and are there other equations like it?" As it turns out, there are lots of ways to understand the special structure of the KdV equation, and there are lots of other equations with the same properties. As we discover them, study them, solve them and find applications for them in the real world, we call all of these results "soliton theory".
Structure of KdV	The incredibly rich structure of the KdV equation can be understood from many different perspectives, like the many facets of a beautiful gem. In some way, they reflect some common property, but they are really independent and can at least sometimes be found in isolation. Understanding their relationship to each other is one goal of soliton theory that has already helped us to better understand mathematics itself. To get an idea, look at the "methods of solution" below.
Methods to "Solve" KdV	Here is a (partial) list of methods to produce solutions to the KdV equation. I am certain that some experts would disagree with my choices, since I may have left off some important methods (e.g. "prolongation") and also have listed separately some closely related ideas (e.g. Darboux vs. Backlund Transformations). But, the point is just to illustrate the diversity of methods available. I have put a "()" at the end of those that I am hoping to be able to introduce in this class.* Analysis Inverse Scattering Riemann-Hilbert problems Painleve Property Algebra Lax pairs () Rank one conditions () Differential Algebra / Darboux Transformation () Symmetry Methods () Backlund Transformations Loop Group Factorizations Algebraic Geometry Grassmannian Manifolds / Plücker Relations () Algebraic Curves () Burchnall-Chaundy Theory of Commuting Differential Operators / finite gap potentials Surfaces of constant curvature Direct / Hirota Bilinear Method (tau-functions) () Hamiltonian Systems* Conservation Laws ("integrals of motion") (*) Quadrature Symplectic Geometry / bi-Hamiltonian Structure