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Fakultät für Mathematik

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Institute

Institut für Analysis

Arbeitsgruppe Nichtlineare Partielle Differentialgleichungen

Travelling Waves

Travelling Waves

Travelling Waves (Wintersemester 2015/16)

Dozent*in: JProf. Dr. Jens Rottmann-Matthes
Veranstaltungen: Vorlesung (0105400), Übung (0105500)
Semesterwochenstunden: 3+1

Termine
Vorlesung:	Dienstag 15:45-17:15	SR 3.68
	Donnerstag 15:45-17:15 (14-tägig)	SR 3.68
Übung:	Donnerstag 15:45-17:15 (14-tägig)	SR 3.68

Lehrende
Dozent	JProf. Dr. Jens Rottmann-Matthes
	Sprechstunde: Keine mehr
	Zimmer - Kollegiengebäude Mathematik (20.30)
	Email: marion.ewald@kit.edu
Übungsleiter	M.Sc. Robin Braun (Stipendiat)
	Sprechstunde: Montags, 14:30 - 15:30 Uhr und nach Vereinbarung
	Zimmer 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
$u_t=u_{xx}+f(u)$ .

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
phase:
$\lim_{t\to\infty} u(\cdot,t)-U(\varphi_\infty-ct)=0,$
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.

Exercises

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

Literaturhinweise

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.