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.