# Nonlinear Waves 19th June, 2013 - Bielefeld University

## Organizers

• Wolf-Jürgen Beyn
• Sebastian Herr

The workshop is part of the conference program of the DFG-funded SFB 701 "Spectral Structures and Topological Methods in Mathematics" at Bielefeld University.

## Program

All talks will take place in V2-210/216

• 09:30-10:15
Yuri Latushkin (Columbia, MO)

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 well-controlled 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 FitzHugh-Nagumo system. We also relate the contour method to the well-known 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. Rottmann-Matthes.
• 10:15-10:45
Denny Otten (Bielefeld)

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 Ornstein-Uhlenbeck type with an unbounded drift term defined by a skew-symmetric matrix $S\in\mathbb{R}^{d,d}$. Equations of this form determine the shape and angular speed of rotating waves in time-dependent 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 quintic-cubic complex Ginzburg-Landau equation. This study is motivated by the stability problem for rotating waves in several variables.
• 11:15-11:45
Nils Strunk (Bielefeld)

For the $L^2$-subcritical gKdV equation it is well-known that all solutions in $H^1$ are global and bounded in time. For the $L^2$-critical and $L^2$-supercritical gKdV equation blow-up in finite time is expected. Indeed Martel and Merle (2002) proved that there are solutions which blow-up in finite time. In 2012 Koch and Steinerberger proved self-similar blow-up for the slightly supercritical gKdV equation. We will discuss a numerical simulation of the self-similar blow-up for the $L^2$-supercritical gKdV equation, which allows to find numerical solutions even on a large supercritical range.
• 11:45-12:15
Stefan Steinerberger (Bonn)

We study the behavior of the (one-dimensional, real-valued) 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.
• 14:00-14:30
Sebastian Herr (Bielefeld)

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 local-wellposedness and ill-posedness results which have been obained recently in collaboration with Enno Lenzmann.
• 14:30-15:15
Richard Kollar (Bratislava)

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 Evans-Krein function, allows to calculate the Krein signature of an eigenvalue at almost no additional computational cost. The extension is based on the equivalent formulation---the 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 Hamiltonian-Hopf bifurcations.
• 15:45-16:15
Jens Rottmann-Matthes (Bielefeld)

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, non-strictly hyperbolic reaction diffusion systems. These naturally arise, if instead of Fouriers law one uses the Cattaneo-Maxwell law as a constitutive relation for the flux of reactants.
• 16:15-16:45
Wolf-Jürgen Beyn (Bielefeld)

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.