Workgroup Nonlinear Partial Differential Equations

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Kollegiengebäude Mathematik (20.30)
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Karlsruher Institut für Technologie
Institut für Analysis
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# Travelling Waves (Summer Semester 2014)

 Lecturer: JProf Dr. Jens Rottmann-Matthes Lecture (0155200), Problem class (0155300) 3+1

In this lecture we consider travelling wave solutions of time-dependent partial differential equations posed on the real line. These are special solutions of the simple form ,
where is a fixed profile of the wave and is its speed.

The easiest example, supporting such a solutions, is the linear transport equation in But these kind of solutions arise also in many other and, in particular, nonlinear problems. One of the most famous equations is the spatially extended Hodgkin-Huxley system, modelling the travelling of pulses along axons.

 Lecture: Problem class: Tuesday 15:45-17:15 Z 2 Thursday 11:30-13:00 Z 2
 Lecturer JProf Dr. Jens Rottmann-Matthes Office hours: - Room - Kollegiengebäude Mathematik (20.30) Email: marion.ewald@kit.edu M.Sc. Robin Braun (Scholarship holder) Office hours: Tuesday, 10:00 - 11:00 and by appointment Room 3.031 Kollegiengebäude Mathematik (20.30) Email: Robin.Flohr@kit.edu

We first consider several equations that give rise to travelling wave solutions.

Then we will look at the large class of reaction-diffusion equations, which model many important phenomena, for example in biological systems. A main emphasis will be on the stability of travelling wave solutions. I.e. the time asymptotic behaviour of solutions with initial values close to the profile of a travelling wave. We will see that the stability is closely related to spectral properties of certain operators, obtained by linearization about the profile of the wave.

Therefore, we will then consider the spectral properties of such operators. A major difficulty is actually to locate the point spectrum and we introduce the Evans-function, which is an important tool for this. Since it is often not possible to analytically calculate the spectrum, we will also look at numerical methods that are well suited to approximate the spectrum.

In the final part of the lecture, we will also have a brief look at travelling waves in Hamiltonian systems.

# Exercises

 Exercise sheet 1 Exercise sheet 2 Exercise sheet 3 Exercise sheet 4 Exercise sheet 5 Exercise sheet 6

# References

The list of references will be updated during the lecture.

### Review Articles and Books

• T. Kapitula, K. Promislow: Spectral and dynamical stability of nonlinear waves, Springer 2013
• R. Knobel: An introduction to the mathematical theory of waves, AMS 2000
• B. Sandstede: Stability of travelling waves, in Handbook of dynamical systems Vol. 2, pp. 983-1055, North-Holland 2002
• A.I. Volpert, V.A. Volpert, V.A. Volpert: Traveling wave solutions of parabolic systems, Translations of Mathematical Monographs, Volume 140, AMS 1994
• W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander: Vector-valued Laplace transforms and Cauchy problems, 2nd ed., Birkhäuser 2011

### Original Research Articles

• D.H. Sattinger: On the stability of waves of nonlinear parabolic systems, Advances in Math., 22(3):312--355, 1976
• M. Grillakis, J. Shatah, W. Strauss: Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal., 74:160--197, 1987
• D.G. Aronson, H.F. Weinberger: Nonlinear Diffusion in Population Genetics, Combustion, and Nerve Pulse Propagation, Lect. Notes Math., 446:5--49, 1975
• D.G. Aronson, H.F. Weinberger: Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30:33--76, 1978