Interactions between Dynamics and PDE

The course provides an introduction to results and methods at the interface between dynamical systems and partial differential equations. In particular, basic ideas will be covered to transfer techniques from ordinary to partial differential equations. Several concrete classes of partial differential equations arising in applications in the natural and engineering sciences will covered. A focus will be on dynamical systems aspects, i.e., qualitative and quantitative properties of solutions will be discussed. Planned topics are:
• bifurcation theory (Lyapunov-Schmidt reduction, local bifurcations),
• travelling waves (existence, stability, propagation speed),
• pattern formation (amplitude equations),
• geometric theory (pattern formation, spiral waves),
• potential further topics (conservation laws, spike solutions, blow-up).
The choice of further topics will be adapted to the interest and background of the audience. In particular, we shall try to incorporate additional dynamical systems tools that could be helpful for a master’s thesis or for research projects if the audience is particularly interested in these directions.

After successful completion of the module students are able to understand and apply the basic notions, concepts, and methods of dynamical systems theory to the analysis of basic classes of partial differential equations. They master in particular the use of elements of Lyapunov-Schmidt reduction, the analysis of travelling waves as well as derivation and use of modulation equations. They know fundamentals of invariant manifold theory and can apply the theoretical results to determine the existence and stability of stationary and non-stationary solutions of partial differential equations.

At least a first course in ordinary differential equations (e.g. MA2005) and partial differential equations (e.g. MA3005).

Lecture (3SWS) + Exercises (1SWS); lectures with blackboard are designed to be interactive and students are encouraged to actively participate both by asking questions as well as by answering questions asked by the instructor. In the exercise classes, students discuss under instructor assistance the results of homework assignments.

blackbaord

Lecture notes will be provided.

The exam will be written or oral depending on the number of students register for the exam during the year the course is given (more than 15 students implies a written exam at 60 minutes, otherwise there will be oral exams of about 25-30 minutes per student). The main requirement for the exam is to have a solid understanding of the underlying theory of dynamical systems methods applied to partial differential equations and vice versa. Students should be able to (a) have a solid knowledge of basic results ('reproduction') (b) be able to apply the results to particular PDEs ('application') and (c) have an understanding of the differences and similarities of some ODE and PDE methods ('transfer'/'comparison'). The exam is going to check the parts (a)-(c).