Bachelor-, prosjekt- og masteroppgaver for Katrin Grunert

My research focuses on nonlinear partial differential equations that govern the motion of waves. In particular, I am interested in wave phenomena and related stability questions.

Wave phenomena: Differential equations describing the motion of waves should also model wave phenomena such as wave breaking. One possible question is therefore: Given a wave profile, can one predict whether or not wave breaking can be observed?

In the case of water waves close to a shore, one has an intuitive understanding of what wave breaking means. As we know from experience, if wave breaking occurs, a lot of energy concentrates in a single point for a moment. But some of the energy is going to disappear immediately afterwards and hence affects the future shape of the wave. Thus, dependent on how much energy is lost one obtains different future shapes of the wave. To obtain a unique continuation, one has to impose additional constrains.

To illustrate what one means by non-uniqueness, consider the following initial value problem, \[\dot x(t)= \sqrt{\vert x(t)\vert },\quad x(0)=a, \quad (a\in\mathbb{R}),\] which does not have a unique global solution. In fact there are infinitely many solutions to the above problem. Introducing some additional constraints one can single out a unique global solution.

When studying the motion of waves, the above mentioned additional constrains have an influence on the future shape of the wave. This means also that they have an influence on how to compare two waves with respect to time. A central question is therefore: Which influence has a small change to the wave profile on the future shape of the wave?

As an illustration, watch the following two movies of solutions to the so-called Camassa—Holm equation, which describes shallow water waves:

Initially, each of them has one peak to the left and an antipeak to the right. Until wave breaking takes place, the peak moves to the right, while the antipeak travels to the left. The first movie shows, what happens if all the energy is lost upon wave breaking: The wave profiles get closer and closer until they coincide. In the second movie, no energy is lost and the behavior changes drastically. The peak and the antipeak pass through each other and the difference between the two wave profiles increases after wave breaking. The shape of the two waves, on the other hand, remains quite similar. Therefore, it is more "natural" to compare shapes in this case.

Studying the influence of small changes to the wave profile on the future shape of the wave plays also an important role from a numerical point of view, since a numerical method should exploit the intrinsic properties of the type of solution of interest.

Bachelor thesis topics: Hvis du er interessert i å skrive en bachelor oppgave, ta kontakt for å avtale et møte, slik at vi kan finne et tema som du har lyst å jobbe med og som er bra egnet for en bachelor oppgave.

Possible master project and thesis topics are within hyperbolic equations and can be about

  • characterising different types of unique (weak) solutions:
    • understanding why weak solutions are not unique and how they can be continued
    • which constraints are needed to single out a unique global solution
  • investigating the stability of such solutions:
    • understanding which small changes in the initial data guarantee that the solution is only slightly influenced
    • identifying ways of comparing solutions
    • understanding the dependence of the solution on the chosen solution concept
  • setting up a numerical method:
    • for a specific type of solutions
    • for comparing different types of solutions with the same initial data

Note: I am far from being an expert in numerics. Therefore a project or thesis concerning numerical methods will be combining analysis and numerics. Emphasis will be put on: Do the numerical results reflect what the analysis predicts and the other way round?

Note: Possible project and thesis topics are not restricted to hyperbolic equations, also topics about ODEs are possible.

Recommended prerequisites for a master thesis (but not a must):

In case of interest, write an email or stop by my office, to discuss possible topics based on your interests and your mathematical background.

2023-11-03, Katrin Grunert