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C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a MAS Modelling, Analysis and Simulation Modelling, Analysis and SimulationThe dynamics of modulated wave trains A. Doelman, B. Sandstede, A. Scheel, G. Schneider The dynamics of modulated wave trains ABSTRACT We investigate the dynamics of weakly-modulated nonlinear wave trains. For reaction-diffusion systems and for the complex Ginzburg--Landau equation, we establish rigorously that slowly varying modulations of wave trains are well approximated by solutions to Burgers equation over the natural time scale. In addition to the validity of Burgers equation, we show that the viscous shock profiles in Burgers equation for the wave number can be found as genuine modulated waves in the underlying reaction-diffusion system. In other words, we establish the existence and stability of waves that are time-periodic in appropriately moving coordinate frames which separate regions in physical space that are occupied by wave trains of different, but almost identical, wave number. The speed of these shocks is determined by the Rankine--Hugoniot condition where the flux is given by the nonlinear dispersion relation of the wave trains. The group velocities of the wave trains in a frame moving with the interface are directed toward the interface. Using pulse-interaction theory, we also consider similar shock profiles for wave trains with large wave number, that is, for an infinite sequence of widely separated pulses. The results represented here are then applied to the FitzHugh--Nagumo equation and to hydrodynamic stability problems. REPORT MAS-E0504 JANUARY 2005 AbstractWe investigate the dynamics of weakly-modulated nonlinear wave trains. For reaction-diffusion systems and for the complex Ginzburg-Landau equation, we establish rigorously that slowly varying modulations of wave trains are well approximated by solutions to Burgers equation over the natural time scale. In addition to the validity of Burgers equation, we show that the viscous shock profiles in Burgers equation for the wave number can be found as genuine modulated waves in the underlying reactiondiffusion system. In other words, we establish the existence and stability of waves that are time-periodic in appropriately moving coordinate frames which separate regions in physical space that are occupied by wave trains of different, but almost identical, wave number. The speed of these shocks is determined by the Rankine-Hugoniot condition where the flux is given by the nonlinear dispersion relation of the wave trains. The group velocities of the wave trains in a frame moving with the interface are directed toward the interface. Using pulse-interaction theory, we also consider similar shock profiles for wave trains with large wave number, that is, for an infinite sequence of widely separated pulses. The results represented here are then applied to the FitzHugh-Nagumo equation and to hydrodynamic stability problems.

Abstract. We investigate, in a systematic fashion, coherent structures, or defects, which serve as interfaces between wave trains with possibly different wavenumbers in reaction-diffusion systems. We propose a classification of defects into four different defect classes which have all been observed experimentally. The characteristic distinguishing these classes is the sign of the group velocities of the wave trains to either side of the defect, measured relative to the speed of the defect. Using a spatial-dynamics description in which defects correspond to homoclinic and heteroclinic connections of an ill-posed pseudoelliptic equation, we then relate robustness properties of defects to their spectral stability properties. Last, we illustrate that all four types of defects occur in the one-dimensional cubicquintic Ginzburg-Landau equation as a perturbation of the phase-slip vortex.

A continuation approach to the computation of essential and absolute spectra of differential operators on the real line is presented. The advantages of this approach, compared with direct eigenvalue computations for the discretized operator, are the efficient and accurate computation of selected parts of the spectrum (typically those near the imaginary axis) and the option to compute nonlinear travelling waves and selected eigenvalues or other stability indicators simultaneously in order to locate accurately the onset to instability. We also discuss the implementation and usage of this approach with the software package AUTO and provide example computations for the FitzHugh-Nagumo and the complex Ginzburg-Landau equation.

Modulated travelling waves are solutions to reaction-diffusion equations that are time-periodic in an appropriate moving coordinate frame. They may arise through Hopf bifurcations or essential instabilities from pulses or fronts. In this article, a framework for the stability analysis of such solutions is presented: the reaction-diffusion equation is cast as an ill-posed elliptic dynamical system in the spatial variable acting upon time-periodic functions. Using this formulation, points in the resolvent set, the point spectrum, and the essential spectrum of the linearization about a modulated travelling wave are related to the existence of exponential dichotomies on appropriate intervals for the associated spatial elliptic eigenvalue problem. Fredholm properties of the linearized operator are characterized by a relative Morse-Floer index of the elliptic equation. These results are proved without assumptions on the asymptotic shape of the wave. Analogous results are true for the spectra of travelling waves to parabolic equations on unbounded cylinders. As an application, we study the existence and stability of modulated spatially-periodic patterns with long-wavelength that accompany modulated pulses.

In applications, solitary-wave solutions of semilinear elliptic equationsin infinite cylinders frequently arise as travelling waves of parabolic equations. As such, their bifurcations are an interesting issue. Interpreting elliptic equations on infinite cylinders as dynamical systems in x has proved very useful. Still, there are major obstacles in obtaining, for instance, bifurcation results similar to those for ordinary differential equations. In this article, persistence and continuation of exponential dichotomies for linear elliptic equations is proved. With this technique at hands, Lyapunov Schmidt reduction near solitary waves can be applied. As an example, existence of shift dynamics near solitary waves is shown if a perturbation +h(x, u, {u) periodic in x is added.1997 Academic Press

An equivariant center-manifold reduction near relative equilibria of G-equivariant semiflows on Banach spaces is presented. In contrast to previous results, the Lie group G is possibly non-compact. Moreover, it is not required that G induces a strongly continuous group action on the underlying function space. In fact, G may act discontinuously. The results are applied to bifurcations of stable patterns arising in reaction diffusion systems on the plane or in three-space modeling chemical systems such as catalysis on platinum surfaces and Belousov Zhabotinsky reactions. These systems are equivariant under the Euclidean symmetry group. Hopf bifurcations from rigidly-rotating spiral waves to meandering or drifting waves and from twisted scroll rings are investigated.1997 Academic Press

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