Background: The KdV equation is a famous nonlinear partial differential equation. It is most famous as a model of water waves and for its soliton solutions such as the one shown in this animation:
(For more information about solitons, click here.) Less well-known are its rational solutions, but those are of interest also both because they are simple to describe and because they have a surprising connection to particle dynamics.
Research Problem: Recently there has been interest in the generalization of soliton equations to the case in which the function is matrix-valued. There have been many papers on the matrix generalizations of solitons. However, nobody seems to have carefully studied the rational solutions to the matrix analogue of the KdV equation. In summer 2015, an undergraduate student and I developed some of the tools that would be needed to study them and proved a few basic theorems about them, but we really need to make a few more discoveries and prove a few more theorems before the work is ready to be submitted for publication in a mathematical physics journal. In particular, although we have been able to produce some interesting rational solutions, it would be ideal if we could have a complete characterization of all of them as we have in the scalar case. Another question that remain open is whether the particle dynamics interpretation generalizes to all of the matrix rational solutions.
What Courses/Skills Do I Need To Have Taken? MATH 221 (Calculus III) and MATH 203 (Linear Algebra) along with some computer programming experience would be enough to work on this project.
When Can I Work on the Project? I would be able to work on this in Summer 2016.
Is Funding Available? There has been no grant funding set aside for this project, however there may be grants that we can apply for once we formalize a project.
More Information: Please email Prof. Kasman if you are interested in learning more.