Dynamics of Patterns

This workshop focused on the dynamics of nonlinear waves and spatio-temporal patterns, which arise in functional and partial differential equations. Among the outstanding problems in this area are the dynamical selection of patterns, gaining a theoretical understanding of transient dynamics, the nonlinear stability of patterns in unbounded domains, and the development of efficient numerical techniques to capture specific dynamical effects

The large core limit of spiral waves in excitable media: A numerical approach

We modify the freezing method introduced by Beyn & Thuemmler, 2004, for analyzing rigidly rotating spiral waves in excitable media. The proposed method is designed to stably determine the rotation frequency and the core radius of rotating spirals, as well as the approximate shape of spiral waves in unbounded domains. In particular, we introduce spiral wave boundary conditions based on geometric approximations of spiral wave solutions by Archimedean spirals and by involutes of circles. We further propose a simple implementation of boundary conditions for the case when the inhibitor is non-diffusive, a case which had previously caused spurious oscillations. We then utilize the method to numerically analyze the large core limit. The proposed method allows us to investigate the case close to criticality where spiral waves acquire infinite core radius and zero rotation frequency, before they begin to develop into retracting fingers. We confirm the linear scaling regime of a drift bifurcation for the rotation frequency and the core radius of spiral wave solutions close to criticality. This regime is unattainable with conventional numerical methods.Comment: 32 pages, 17 figures, as accepted by SIAM Journal on Applied Dynamical Systems on 20/03/1

Freezing traveling and rotatingwaves in second order evolution equations

In this paper we investigate the implementation of the so-called freezing method for second order wave equations in one and several space dimensions. The method converts the given PDE into a partial differential algebraic equation which is then solved numerically. The reformulation aims at separating the motion of a solution into a co-moving frame and a profile which varies as little as possible. Numerical examples demonstrate the feasability of this approach for semilinear wave equations with sufficient damping. We treat the case of a traveling wave in one space dimension and of a rotating wave in two space dimensions. In addition, we investigate in arbitrary space dimensions the point spectrum and the essential spectrum of operators obtained by linearizing about the profile, and we indicate the consequences for the nonlinear stability of the wave

Dynamics of patterns in equivariant Hamiltonian partial differential equations

Dieckmann S. Dynamics of patterns in equivariant Hamiltonian partial differential equations. Bielefeld: Universität Bielefeld; 2017

Freezing multipulses and multifronts

Beyn W-J, Selle S, Thümmler V. Freezing multipulses and multifronts. SIAM Journal on Applied Dynamical Systems. 2008;7(2):577-608.We consider nonlinear time dependent reaction diffusion systems in one space dimension that exhibit multiple pulses or multiple fronts. In an earlier paper two of the authors developed the freezing method that allows us to compute a moving coordinate frame in which, for example, a traveling wave becomes stationary. In this paper we extend the method to handle multifronts and multipulses traveling at different speeds. The solution of the Cauchy problem is decomposed into a finite number of single waves, each of which has its own moving coordinate system. The single solutions satisfy a system of partial differential algebraic equations coupled by nonlinear and nonlocal terms. Applications are provided to the Nagumo and the FitzHugh-Nagumo systems. We justify the method by showing that finitely many traveling waves, when patched together in an appropriate way, solve the coupled system in an asymptotic sense. The method is generalized to equivariant evolution equations and is illustrated by the complex Ginzburg-Landau equation