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

# 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