**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.) The soliton solutions are singular limits of the periodic solutions that can often be seen manifested as ocean waves. Less well-known are its *rational* solutions, but those are of interest also because they are simple to describe, because they have a symmetry called “bispectrality, 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.)

In Summer 2018, a team of three students and I studied the rational, solitonic, and periodic solutions to KdV that take values in the quaternions. We found some very interesting results, but there are still open questions.

**Research Problem: **What is the behavior of the nonlinear superposition of different quaternion-valued periodic solutions to the KdV equation? Can the quaternion-valued rational solutions always be written as a sum of square inverses of linear terms (as in the commutative case)? What is the connection between the quaternion-valued rational solutions and particle systems of Calogero-Moser type? And, precisely which rational Lax operators are bispectral?

**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 project in Summer 2019 or later.

**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.