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Aspects of Nonlinear Wave Equations (Summer Semester 2016)

Nonlinear wave equations occur in many mechanical and electromagnetic models. This course is meant to describe some typical phenomena in nonlinear wave equations like the formation of stable traveling or standing waves. The course will be more of exemplary nature rather than of comprehensive nature. I will use many tools and notions from linear and nonlinear analysis, e.g., Sobolev spaces, spectral theory, variational techniques, notions from nonlinear functional analysis like Frechet-differentiablity, implicit function theorem. They will be mostly introduced and explained during the course. The course is meant for advanced Master students. Familiarity with partial differential equations and some functional analysis is indispensable.

Current Evaluation

exercise classes

Notice the following swap:
08.06.2016, 14:00 - 15:30 - lecture
22.06.2016, 11:30 - 13:00 - exercise class

Lecture: Monday 9:45-11:15 SR 3.68
Wednesday 11:30-13:00 SR 3.68
Problem class: Wednesday 14:00-15:30 SR 3.68 Begin: 27.4.2016
Lecturer Prof. Dr. Wolfgang Reichel
Office hours: Monday, 11:30-13:00 Before you e-mail: call or come!
Room 3.035 Kollegiengebäude Mathematik (20.30)
Email: Wolfgang.Reichel@kit.edu
Problem classes M.Sc. Piotr Idzik
Office hours: by appointment
Room 3.038 Kollegiengebäude Mathematik (20.30)
Email: vil02@o2.pl

The questions that I will address are:

  1. Worm-up on the linear wave equation
  2. Existence of traveling waves for u_{tt}-u_{xx}=f(u)
  3. Existence of traveling waves in a suspension bridge model u_{tt}+u_{xxxx} = f(u)
  4. Variational approach to standing, time-periodic waves for u_{tt}-u_{xx}= -|u|^{p-1}u
  5. Stability questions for nonlinear wave equations

Here is a summary of the topics of this lecture (1st version of July 18, 2016).

Problem sheets

Exercise Sheet 1 (corrected version, 22.04.2016) Solutions 1
Exercise Sheet 2 Solutions 2
Exercise Sheet 3 (corrected version, 11.05.2016) Solutions 3
Exercise Sheet 4 Solutions 4
Exercise Sheet 5 Solutions 5 Some remarks
Exercise Sheet 6 Solutions 6
Exercise Sheet 7 (corrected version, 06.06.2016) Solutions 7
Exercise Sheet 8 (corrected version, 21.06.2016) Solutions 8
Exercise Sheet 9 Solutions 9
Exercise Sheet 10 Solutions 10
Exercise Sheet 11 Solutions 11
Exercise Sheet 12 Solutions 12


This is a 4h course with 8 ECTS. The examination will be via an oral exam.

Exam dates: 08.09.2016 (Thursday), 07.10.2016 (Friday)


Among others I will use the following sources (the list will be completed during the course):

  • Adams, Fournier: Sobolev spaces (Elsevier, 2002)
  • Struwe: Variational Methods (Springer, 1996), Chapter I.6
  • Grillakis, Shatah, Strauss: Stability Theory of Solitary Waves in the Presence of Symmetry, Journal of Functional Analysis 74, 160--197 (1987)