Advanced Methods in Nonlinear PDEs (Winter Semester 2024/25)
- Lecturer: Dr. Björn de Rijk
- Classes: Lecture (0100002)
- Weekly hours: 2
| Schedule | ||
|---|---|---|
| Lecture: | Wednesday 9:45-11:15 | 20.30 SR 2.58 |
| Lecturers | ||
|---|---|---|
| Lecturer | Dr. Björn de Rijk | |
| Office hours: Office hours: by appointment | ||
| Room -1.019 Kollegiengebäude Mathematik (20.30) | ||
| Email: bjoern.rijk@kit.edu | ||
Contents
Nonlinear partial differential equations describing physical phenomena are often complex, making their qualitative and quantitative analysis challenging. Amplitude or modulation equations, such as the Ginzburg-Landau equation, the Korteweg-de Vries equation, and the nonlinear Schrödinger equation, play an important role in capturing the critical dynamics of spatially extended dissipative or conservative physical models. Mathematical theorems demonstrate that these well-understood asymptotic models accurately predict the behavior of the original system on sufficiently long time scales. Examples of regimes which can be described in such a way include pattern-forming systems close to their first instability, the long-wave limit of the water wave problem, and highly oscillatory regimes in nonlinear optics.
In the first part of this course, we develop several methods to rigorously justify approximations of complex physical systems by amplitude or modulation equations. Relevant tools include Fourier analysis, energy estimates, semigroup theory, mode filters, and normal form transformations. Often, amplitude or modulation equations admit special solutions, such as periodic patterns, solitary waves, or traveling (modulating) fronts. While approximation results yield solutions of the original system that are close to these special solutions, they are insufficient to conclude that such special solutions exist in the original system as well. In the second part of this course, we focus on techniques, such as Lyapunov-Schmidt reduction, spatial dynamics, and center manifold reduction, to construct these special solutions in the original system.
Examination
The module examination at the end of the semester takes place in the form of an oral exam of about 30 minutes.
References
We will roughly follow Chapter 10-13 from the book:
- Nonlinear PDEs: A Dynamical Systems Approach by Guido Schneider and Hannes Uecker