Nonlinear Waves
19th June, 2013  Bielefeld University
Organizers
 WolfJürgen Beyn
 Sebastian Herr
The workshop is part of the conference program of the DFGfunded
SFB 701 "Spectral Structures and Topological Methods in Mathematics" at Bielefeld University.
Program
All talks will take place in V2210/216

09:3010:15
Yuri Latushkin (Columbia, MO)
Nonlinear Eigenvalues, Keldysh's Theorem, and the Evans function
We propose to detect and approximate the point spectra of differential operators (and the associated eigenfunctions) via contour integrals of solutions to resolvent equations. The approach is based on Keldysh' theorem and extends a recent method for matrices depending analytically on the eigenvalue parameter. We show that errors are wellcontrolled under very general assumptions when the resolvent equations are solved via boundary value problems on finite domains. Two applications are presented: an analytical study of Schroedinger operators on the real line as well as on bounded intervals and a numerical study of the FitzHughNagumo system. We also relate the contour method to the wellknown Evans function and show that our approach provides an alternative to evaluating and computing its zeros. This is a joint work with W.J. Beyn and J. RottmannMatthes.

10:1510:45
Denny Otten (Bielefeld)
Spatial decay of rotating waves in parabolic systems
In this talk we study solutions of nonlinear systems
$A\triangle v(x)+\left\langle Sx,\nabla v(x)\right\rangle+f(v(x))=0,\,x\in\mathbb{R}^d,\,d\geqslant 2$.
The linear operator is of OrnsteinUhlenbeck type with an unbounded drift term defined by a
skewsymmetric matrix $S\in\mathbb{R}^{d,d}$. Equations of this form determine the shape and angular
speed of rotating waves in timedependent reaction diffusion systems. We prove under certain
conditions that every classical solution which falls below a certain threshold at infinity,
must decay exponentially in space. Examples for such solutions are spinning solitons arising
from the quinticcubic complex GinzburgLandau equation. This study is motivated by the stability
problem for rotating waves in several variables.

11:1511:45
Nils Strunk (Bielefeld)
Numerical Simulation of Selfsimilar Blowup of the $L^2$supercritical gKdV equation
For the $L^2$subcritical gKdV equation it is wellknown that all
solutions in $H^1$ are global and bounded in time. For the
$L^2$critical and $L^2$supercritical gKdV equation blowup in finite
time is expected. Indeed Martel and Merle (2002) proved that there
are solutions which blowup in finite time. In 2012 Koch and
Steinerberger proved selfsimilar blowup for the slightly
supercritical gKdV equation. We will discuss a numerical simulation of
the selfsimilar blowup for the $L^2$supercritical gKdV equation, which allows to find numerical solutions even on a large supercritical range.

11:4512:15
Stefan Steinerberger (Bonn)
Dispersion dynamics for the defocusing generalized Kortewegde Vries equation
We study the behavior of the (onedimensional, realvalued) defocusing gKdV equation. It is expected that it is not possible for most of the $L^2$mass (which is conserved) to concentrate in a small
spatial interval for a long time. We survey the existing results by Tao and Kwon & Shao and give an improved estimate.

14:0014:30
Sebastian Herr (Bielefeld)
The Boson Star Equation for Rough Data
We will discuss a nonlinear dispersive initial value problem
which arises in the effective description of the dynamics and
gravitational collapse of boson stars. We will present essentially
optimal localwellposedness and illposedness results which have been
obained recently in collaboration with Enno Lenzmann.

14:3015:15
Richard Kollar (Bratislava)
Graphical Krein signature and its applications
The key question in spectral stability of nonlinear waves is the existence of eigenvalues with positive real part of a linearized operator. Two concepts very different in nature proved to be useful in search for such an unstable spectrum: the Evans function, an analytic function with zeros at isolated unstable eigenvalues, and the Krein signature, an algebraic quantity capturing the ability of an eigenvalue to be or to become unstable under a change of a parameter in a system. Although the Evans function does not provide full information about the Krein signature, we show that its simple extension, the EvansKrein function, allows to calculate the Krein signature of an eigenvalue at almost no additional computational cost. The extension is based on the equivalent formulationthe graphical Krein signature. The method used also enables us to give very elegant proofs of index theorems for linearized Hamiltonians and general eigenvalue pencils and it also leads to a new class of sufficient criteria for HamiltonianHopf bifurcations.

15:4516:15
Jens RottmannMatthes (Bielefeld)
Freezing traveling waves with application to a hyperbolic HodgkinHuxley system
An important step in understanding the solution of systems of nonlinear
partial differential equations is the analysis of relative equilibria like
traveling waves (one spatial dimension) or rotating waves (two spatial
dimensions). In this talk I will mainly focus on how to capture traveling
waves numerically by using the freezing method. Generalizations to more
general symmetries will also be considered.
As an example class I look at the case of semilinear, nonstrictly
hyperbolic reaction diffusion systems. These naturally arise, if instead
of Fouriers law one uses the CattaneoMaxwell law as a constitutive
relation for the flux of reactants.

16:1516:45
WolfJürgen Beyn (Bielefeld)
Stability and computation of interacting nonlinear waves
In dissipative evolution equations one often observes special
solutions that are composed of several nonlinear waves. If these
waves travel at different speeds they may either collide
(strong interaction) or repel each other (weak interaction).
While some theory is currently emerging for the case of weak interaction, there
is hardly any rigorous result about the fascinating phenomena
during strong interaction.
In this talk we report about an extension of the freezing method
which allows to compute multiple coordinate frames. In these frames,
single profiles can stabilize independently while still capturing
their nonlinear interaction. The basic idea is to use dynamic
partitions of unity and to decompose the evolution equation
into a system of partial differential algebraic equations the dimension
of which is given by the maximal number of waves.
For the case of weakly interacting traveling waves in one space dimension,
we present an asymptotic stability result for the decomposition system.
The numerical experiments go far beyond this situation and show
how the method copes with weakly and strongly interacting
waves in one and two space dimensions.
This is joint work with Denny Otten and Sabrina Selle.
For further information please contact Mr WolfJürgen Beyn: beyn(at)math(dot)unibielefeld(dot)de or Mr Sebastian Herr: herr(at)math(dot)unibielefeld(dot)de