Background: The general goal of the project is to study equations which describe the behavior of water waves. One of the most well-known equations describing water waves is the Korteweg-de-Vries equation (KdV). However, there are other water wave models and the project will deal with the KdV and another related equation: the Degasperis-Processi.
The Degasperis-Processi can be used to describe the phenomenon of wave breaking, which occurs when the top of the wave starts to crest. Both the models under consideration are differential equations, that is they are equations involving derivatives. The solutions to these equations represent behaviors for the water waves. The goal is to study the stability properties of the solutions. Stability is a fundamental concept, which can be illustrated by trying to make a pencil stand on its lead. Because it is such an unstable state, it is possible in theory only. The concept of stability carries over to differential equations and its study often involves sophisticated mathematical tools.
Research Problems: The main goal of the project is to develop methods to determine whether a given solution is stable. While everything is known about the stability of the solutions of the KdV equation, it is not the case for the Degasperis-Processi. We plan to use the KdV equation as a way for the student to learn the general methods. Once this is done, we will tackle the problem of studying the stability of the solutions for the Degasperis-Processi.
What Courses/Skills Do I Need To Have Taken? MATH 323 (Differential Equations).
When Can I Work on the Project? The work would be started in the Summer but could be continued during the Fall and Spring semesters.
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.