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.
The quaternions are a generalization of the real and complex numbers first discovered by William Rowan Hamilton in the 19th century. Like the complex numbers, these involve square roots of -1, but the quaternions are four-dimensional (rather than two-dimensional like the complex numbers) and do not satisfy the commutative law. They have a more complicated algebraic structure, which is closely related to the symmetries of physical space. (For more information about quaternions, click here.)
Research Problem: The goal of this project is to study solutions to a non-commutative version of the KdV equation whose solutions are quaternion-valued. I believe we can easily produce such solutions using a procedure analogous to the one used in the real and complex-valued cases, but since nobody has studied these solutions I am not sure what we will find. The project will hopefully involve a combination of experimentation, creating animations to illustrate the dynamics, and rigorously proving results about these novel solutions to non-commutative nonlinear wave equations. From my perspective, one of the most interesting things would be to study the bispectrality of the corresponding Lax operators.
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 Spring 2018 or Summer 2018.
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.