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Travelling Waves (Winter Semester 2015/16)

Lecture: Tuesday 15:45-17:15 SR 3.68
Thursday 15:45-17:15 (every 2nd week) SR 3.68
Problem class: Thursday 15:45-17:15 (every 2nd week) SR 3.68
Lecturer JProf Dr. Jens Rottmann-Matthes
Office hours: -
Room - Kollegiengebäude Mathematik (20.30)
Email: marion.ewald@kit.edu
Problem classes 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

In this lecture we will consider traveling wave solutions to partial
differential equations in 1+1-dimensions. These are solutions
u:\mathbb{R}\times[0,\infty)\to\mathbb{R}^m of the form
u(x,t)=U(x-ct), to a system of reaction-diffusion equations

Here U:\mathbb{R}\to\mathbb{R}^m denotes the profile and c denotes
the velocity of the wave.

First we will consider the question of existence of such solutions.
After that we will look at the question of stability, i.e. whether a
small perturbation of a traveling wave solution converges to the
traveling wave solution as time tends to \infty. Since the equation
is space independent, one can only expect stability with asymptotic
\lim_{t\to\infty} u(\cdot,t)-U(\varphi_\infty-ct)=0,
for some suitable \varphi_\infty.

Good knowledge of Analysis I-III, good knowledge of Linear Algebra, some knowledge of Functional Analysis, some knowledge of Spectral theory, some knowledge of PDEs.

Furthermore, Bochner-Integration might be helpful but is not necessary.


Exercise sheet 1
Exercise sheet 2
Exercise sheet 3
Exercise sheet 4
Exercise sheet 5
Exercise sheet 6
Exercise sheet 7


W. Arendt, Charles J. K. Batty, M. Hieber, and F. Neubrander: Vector-valued Laplace transforms and Cauchy problems. Volume 96 of Monographs in Mathematics, Birkhäuser Verlag, Basel, 2001.

D. G. Aronson and H. F. Weinberger: Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974), Volume 446 of Lecture Notes in Mathematics, pages 5-49. Springer, Berlin, 1975.

L. C. Evans: Partial differential equations. Volume 19 of Graduate Studies in Mathematics, AMS, Providence, RI, 1998.

Roger Knobel: An introduction to the mathematical theory of waves. Providence, RI: American Mathematical Society, 2000.

Heinz-Otto Kreiss and Jens Lorenz: Initial-boundary value problems and the Navier-Stokes equations. Volume 136 of Pure and Applied Mathematics, Academic Press Inc., Boston, MA, 1989.

Björn Sandstede: Stability of travelling waves. Handbook of dynamical systems, Vol. 2, pages 983-1055, North-Holland, Amsterdam, 2002.