**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.