MATHEMATISCHES FORSCHUNGSINSTITUT OBERWOLFACH
Miniworkshop: Dynamics of Nonlinear Waves
January 19th - January 25th, 2003

Abstracts

$\parbox{\textwidth}{ \begin{center}\textbf{Symplectic structure of t... ...nelse{\equal{}{}}{} { \\ [\smallskipamount] (joint work with ) } \end{center}}$
The aim of this talk was to give an informal introduction to the linear stability analysis of solitary wave solutions of Hamiltonian PDEs using the Evans function. The backbone of the analysis is the multi-symplectic representation of Hamiltonian PDEs. The talk began with an introduction to multisymplecticity. Then, the implications of this structure for the existence and linear stability problem was presented. The Evans function was shown to have a geometric representation, in terms of the determinant of a matrix of restricted symplectic forms. A geometric instability condition was presented, and the theory illustrated by application to a 2-parameter family of solitary waves of the Boussinesq equation. An introduction to the implications of multisymplecticity for two (and more) space dimensions was also given. An application of the 2+1-dimensional case is to the transverse instability of solitary waves. Other applications such as symplectic structure of soliton interaction (for example in the Zakharov-Kuznetsov equation), analyticity of the symplectic Evans function, the potential role of the Maslov index, extension to infinite dimensions of the symplectic Evans function, solitary waves of Hamiltonian PDEs on manifolds and other potential open problems were also briefly discussed.

$\parbox{\textwidth}{ \begin{center}\textbf{Destabilization of fronts... ...nelse{\equal{}{}}{} { \\ [\smallskipamount] (joint work with ) } \end{center}}$
We consider a class of bi-stable reaction-diffusion equations in two components on the real line. We assume that the system is singularly perturbed, i.e. that the ratio of the diffusion coefficients is (asymptotically) small. This class admits front solutions that are asymptotically close to the (stable) front solution of the `trivial' scalar bi-stable limit system u_t = u_xx + u(1-u²). However, in the system these fronts can become unstable by varying parameters. This destabilization is either caused by the essential spectrum associated to the linearized stability problem, or by an eigenvalue that exists near the essential spectrum. We use the Evans function to study the various bifurcation mechanisms and establish an explicit connection between the character of the destabilization and the possible appearance of saddle-node bifurcations of heteroclinic orbits in the existence problem.

$\parbox{\textwidth}{ \begin{center}\textbf{Dynamics of pulse-like lo... ...nelse{\equal{}{}}{} { \\ [\smallskipamount] (joint work with ) } \end{center}}$
I propose a new type of Billiard problem in which angles of incidence and reflection are different. This problem comes from the dynamics of pulse-like localized patterns in R-D systems in 2D. In fact, a system of 3-components possesses a moving pulse- like localized solution which moves with constant velocity and constant shape. It is called "a traveling spot solution". The ODE which describes the motion of a traveling spot is given in mathematically rigorous way under some assumptions. Moreover, the case that there exist multi-traveling spot solutions in R² are also treated. As the consequence, they interact repulsively each other. Next, we consider the motion of one traveling spot in some region with Neumann boundary conditions. Then, a traveling spot reflect from the boundary. In the reflection, the angle of incidence is larger than the angle of reflection, which gives a new type of Billiard problem. If the region is a square, there exists a stable limit cycle of orbit. On the other hand, if we consider a rectangle with the ratio L/M and change it, some chaotic orbit which almost fills the region appears. We also give a picture of the global structure of the orbits with respect to the ratio by numerics.

$\parbox{\textwidth}{ \begin{center}\textbf{Stability of modulated fr... ...llskipamount] (joint work with Guido Schneider, Hannes Uecker) } \end{center}}$
When the steady states at infinity become unstable through a pattern forming bifurcation, a travelling wave may bifurcate into a modulated front which is time-periodic in a moving frame. This scenario has been studied by B. Sandstede and A. Scheel for a class of reaction-diffusion systems on the real line. Under general assumptions, they showed that the modulated fronts exist and are spectrally stable near the bifurcation point. I present here a model problem for which one can prove the nonlinear stability of these solutions with respect to small localized perturbations. This result does not follow from the spectral stability, because the linearized operator around the modulated front has essential spectrum up to the imaginary axis.

$\parbox{\textwidth}{ \begin{center}\textbf{Mathematical modelling an... ...nelse{\equal{}{}}{} { \\ [\smallskipamount] (joint work with ) } \end{center}}$
The classical model for subsonic flame fronts in diffusive gaseous mixtures under the assumption of simple chemistry is formulated as a free boundary problem for temperature and fuel concentration. The reaction takes place at the free boundary which is moving into the fresh zone leaving the burnt region behind. Mass flux going into the flame is balanced by heat flux coming out of the flame and is given by a temperature dependent reaction rate.

If the mixture contains dust, a radiation effect has to be taken into account which has a nonlocal effect on the temperature equation. Modelling the radiative flux by the Eddington equation of astrophysics, we find that the flame front temperature is enhanced to values well above the adiabatic temperature. The presence of dust makes the medium more flammable.

This model has been suggested by both Joulin and Buckmaster. Our goal is to obtain a complete bifurcation and stability picture of travelling waves in terms of the temperature and fuel concentration far away ahead of the flame, and the relevant parameters in the model. These are, after some normalisations, the Lewis number, the opacity, the Boltzmann number and the Zel'dovich number.

$\parbox{\textwidth}{ \begin{center}\textbf{A Morse Index Theorem for... ...Deng}{}}{} { \\ [\smallskipamount] (joint work with Jian Deng) } \end{center}}$
It has been a long standing issue to relate the structure of a solution of an elliptic boundary value problem with its degree of instability, i.e. its Morse Index. In one dimension, this is easily achieved using Sturm-Liouville theory which affords a relationship between the number of sign-changes of the derivative of the solution with its Morse Index. A generalization of this idea to systems of elliptic equations with a gradient nonlinearity is known and based on the idea that the linearization of the system generates a flow on the space of Lagrangian planes. This is closely related to the Morse Index Theorem which was originally formulated for geodesics on a Riemannian manifold. We prove a result of a similar structure but for solutions of elliptic problems in arbitrary space dimensions, as long as the underlying domain is start-shaped. The shrinking of the domain supplies the parameter that renders a curve of Lagrangian subspaces from the linearization. The key is to view these subspaces as lying in a Fredholm Lagrangian of infinite-dimensional subspaces and develop a Maslov Index theory. The result generalizes earlier work of Smale to now cover more general boundary conditions and explains why Smale's result only worked for Dirichlet conditions.

$\parbox{\textwidth}{ \begin{center}\textbf{Semi-strong interaction o... ...nelse{\equal{}{}}{} { \\ [\smallskipamount] (joint work with ) } \end{center}}$
Pulse-pulse interactions play central roles in a variety of pattern formation phenomena, including self-replication. In this talk, we present theory for the semi-strong interaction of pulses in a class of singularly perturbed, coupled reaction-diffusion equations that includes the (generalized) Gierer-Meinhardt, Gray-Scott, Schnakenberg, and Thomas models, among others. Geometric conditions are determined on the reaction kinetics for whether the pulses in a two-pulse solution attract or repel, and ordinary differential equations are derived for the time-dependent separation distance between their centers and for their wave speeds. In addition, conditions for the existence of stationary two-pulse solutions are identified, and the interactions between stationary and dynamically-evolving two-pulse solutions are studied. The theoretical results are illustrated on a series of examples. In two of these, which are related to the classical Gierer-Meinhardt equation, we find that the pulse amplitudes blow up in finite time. Moreover, the blowup of stationary one-pulse solutions and of dynamically-varying two-pulse solutions occurs precisely at the parameter values for which the theory we develop predicts that these solutions should cease to exist. Finally, for one of these examples, we discover a new type of codimension two point in which the bifurcation curve of pulse-splitting and the bifurcation curve of blowup intersect.

$\parbox{\textwidth}{ \begin{center}\textbf{Evans' function for infin... ...nelse{\equal{}{}}{} { \\ [\smallskipamount] (joint work with ) } \end{center}}$
The aim of the talk is to illustrate the construction of Evans' function in infinite-dimensional setting.

The classical Evans' function was defined to be the determinant of solutions which decay in plus or minus infinity. Several way of defining Evans' function have been used which depend on some finite dimensionality of each infinite dimensional system.

In this talk, a new definition of Evans' function is introduced, which depends on Fredholm property of the system rather than infinite dimensionality. The function is defined to be an analytic section to the pull back of the determinant bundle over the space of Fredholm operators. The multiplicity lemma and winding number argument can also be shown in general framework.

$\parbox{\textwidth}{ \begin{center}\textbf{Scattering dynamics in di... ...nelse{\equal{}{}}{} { \\ [\smallskipamount] (joint work with ) } \end{center}}$
Scattering of particle-like patterns in dissipative systems is studied, especially we focus on the issue how the input-output relation is controlled at a head-on collision in 1D and 2D spaces where traveling pulses and spots collapse. It remains an open problem due to the large deformation of patterns at a colliding point. We found that special type of steady or time-periodic solutions called separators and their stable and unstable manifolds direct the traffic flow of orbits. Such separators are clearly visible at a transition point where the input-output relation is qualitatively changed such as from annihilation to repulsion as parameters vary. Separators are in general highly unstable which causes a variety of input-output relations during the process of scattering.

$\parbox{\textwidth}{ \begin{center}\textbf{Coherent Manifolds for Da... ...nelse{\equal{}{}}{} { \\ [\smallskipamount] (joint work with ) } \end{center}}$
The interaction of light with matter is dominated by diffraction and dispersion, absorption and stimulated emission all combining to produce a noisy background of radiation. Against this background, certain localized structures display a remarkable coherence through a robust balancing of nonlinearity against forcing and dispersion. The emerging technology of modern optics seeks to control these coherent structures for purposes of data propagation, pattern generation, and optical logic.

From these coherent localized structures we iteratively build up a coherent manifold: a low-dimensional, invariant manifold which captures the salient dynamics of pulse interaction, bifurcation of fixed points and traveling waves, oscillatory instabilities and modulation of multi-dimensional fronts.

The manifold is a sum of a base $\Phi_0$ describing a collection of localized coherent structures, and two small extensions,

$$\Phi(x; {\vec{p}}) = \Phi_0 +\Phi_1 +\Phi_2,$$ (1)

where $\Phi_1$ lies in a finite dimensional subspace and serves to extend the dimension of the tangent plane of the manifold, and $\Phi_2$, the radiative extension, lies in the large, complementary subspace and is chosen to enforce the invariance or quasi-invariance of the manifold, capturing radiative effects arising from the interplay between forcing and damped dispersion. The manifold coordinates ${\vec{p}}$ represent slow modes of the system and are typically identified with localized eigenmodes of the linearization $L_ {\vec{p}}$ of the vector field about $\Phi_0( {\vec{p}})$. These eigenmodes arise, for example, from symmetries or self-similar scalings of the PDE, and are the ``free'' coordinates: position, phase, polarization, which describe the state of the coherent structures.

The construction is based upon

(1)	Detailed analysis of families of linearized operators by
	Evans function and non-perturbative Krein-signature methods
(2)	Development of decay estimates for time-dependent semi-groups
	via renormalization group methods.

As a pro-typical optical system we consider the optical parametric oscillator (OPO) which has attracted recent attention as a source of broadly-tunable coherent radiation for use in pattern recognition technologies, optical information processing, and infrared spectroscopy. OPOs exploit the quadratic $\chi^{(2)}$-nonlinearity of a non-centrosymmetric crystal such as lithium niobate to convert energy parametrically from one frequency to another. The development of quasi-phase matching (QPM) through periodic poling of the crystal's molecular structure has has greatly increased the viability and simplified the control of these devices while extending their operational range and possible applications. In a continuous pumping limit, the OPO process converts a coherent pump field V into a frequency shifted signal field U, governed by the equations in 1+2 dimensions,

$$\begin{array}{lcl} U_t &=& \frac{i}{2} \nabla^2 U+VU^* - (1+... ...i}{2} \rho\nabla^2 V -U^2 - (\alpha+i\Delta_2)V +S, \end{array}$$ (2)

where $\Delta_1$ and $\Delta_2$ are the cavity detuning parameters and S is a spatially dependent pump strength.

We examine pulse-pulse interactions for the OPO system, and describe a snaking-bifurcation which results in multi-pulses. In the second talk we extend the method to consider oscillatory pulses and the generation of stable periodic solutions and address pattern formation in the context of multi-dimensional fronts.

$\parbox{\textwidth}{ \begin{center}\textb... ...nelse{\equal{}{}}{} { \\ [\smallskipamount] (joint work with ) } \end{center}}$
We study the singular limit of a reaction-diffusion system known, in the engineering literature, as the thermo-diffusive model for flame propagation with one-step chemisty. The unknowns are the temperature and the mass fraction of the reactant. There is a diffusivity ratio - the Lewis number, denoted by Le - which is usually taken equal to unity in the mathematical literature on this system. The parameter that makes the system singular is the inverse of the high activation energy $\epsilon$, which makes the reaction term converge - at least formally - to a measure on the surface where the temperature reaches its maximum value as $\epsilon \to 0$.

In the present context, 3D spherical symmetry is assumed, and the surface of maximal temperature is a sphere whose radius (flame radius) has an evolution law to be determined. We prove, under the assumption Le<1, the existence of a class of initial data such that, for the solution starting from these data:

• the correct time scale is $\epsilon^{-2}$,
• the the flame radius, after a rescaling in time, converges as $\epsilon \to 0$ to the solution of a nonlinear integro-differential equation.

Such an equation had been derived by G. Joulin in 1985, but only in the formal style (matched asymptotic expansions). The present paper gives a completely rigorous justification on Joulin's analysis; the method of proof is basically in two steps.

• The construction of an approximate solution to sufficiently high order in $\epsilon$.
• Checking that the true solution is close enough to the approximate one; this involves sharp resolvent and semigroup estimates.

An important open problem is the study of the system on times scales $>>\epsilon$.

$\parbox{\textwidth}{ \begin{center}\textbf{Stability of large Ekman ... ...nelse{\equal{}{}}{} { \\ [\smallskipamount] (joint work with ) } \end{center}}$
We study the limit of the Navier-Stokes equation for an incompressible three dimensional rotating fluid when the Ekman and Rossby number go to zero. Near horizontal boundaries an Ekman layer appears and its stability can be characterized with a Reynolds number. For large Reynolds numbers, the Ekman layer is known to be linearly unstable whereas for very small Reynolds numbers, the stability can be proved by classical energy estimates. Here, we study the stability for intermediate Reynolds numbers by means of Evans function/Symmetrizer's construction techniques.

$\parbox{\textwidth}{ \begin{center}\textbf{Defects in oscillatory m... ...el}{}}{} { \\ [\smallskipamount] (joint work with Arnd Scheel) } \end{center}}$
Among the many patterns that have been observed in experiments and numerical simulations are defects. Defects are modulated waves that are time periodic in an appropriate moving frame and asymptotic in space to wave trains. In this talk, we show that spatial dynamics can be used to analyse defects and their spectral stability properties. It appears as if there are four different types of defects that are robust: sinks, sources, transmission and contact defects. The different defect classes are distinguished by their codimension and the characteristic curves associated with them. Spatial dynamics also allows us to determine the PDE spectrum of defects near the origin in L² and exponentially weighted L² norms. The analysis of the spectrum of contact defects is particularly involved and requires an extension of the Evans function in situations where the Gap Lemma fails. Open problems include a complete classification of defects, the nonlinear stability of sources, transmission and contact defects, bifurcations from and of defects, and finally their interaction properties.

$\parbox{\textwidth}{ \begin{center}\textbf{Extending the Evans funct... ...{ \\ [\smallskipamount] (joint work with Bj\uml orn Sandstede) } \end{center}}$
Contact defects arise in open classes of reaction diffusion systems. Like other types of defects they are time-periodic in a suitably comoving frame and spatially asymptotic to nonlinear wave trains. Their characteristic feature is that the wavenumbers to the left and to the right of the defect coincide, and that the speed of propagation of the defect is given by the group velocity of the wave trains in the far field. We are interested in stability, interaction, and bifurcation properties of these types of defects. We therefore investigate spectral properties of the linearization at the defect, in a neighborhood of the origin, $\lambda=0$. The fact that the speed of propagation is given by the group velocity leads to two difficulties. First, the convergence of the defect to the asymptotic wave trains is only algebraic, $\sim 1/x$, and then $\lambda=0$ is a branch point in the essential spectrum of the defect. In order to characterize spectral properties, we construct an Evans function for the ill-posed modulated wave equation, using first Galerkin approximations globally in $\lambda$, and then center-manifold and Lyapunov-Schmidt reduction, locally near $\lambda=0$. For the actual construction of the Evans function, we use geometric blow-up and resonant normal forms for the differential equation on the reduced Grassmannian. As a main result, we show that the Evans function E can be written as $E(\lambda)=\sqrt{\lambda}E_0(\sqrt{\lambda},\sqrt{\lambda}\log\lambda)$, where $E_0(\cdot,\cdot)$ is $C^\infty$. Moreover, $\partial_2 E_0(0,0)\neq 0$ whenever the decay towards the wavetrains is of the form $1/x+a/x^2+\ldots$ with $a\neq 0$. We also show that a similar construction can be employed to continue the Evans function beyond the Gap Lemma as a meromorphic function. Resonances between the exponential decay of the wave profile and the gap between the leading eigenvalues generate poles in the Evans function.

$\parbox{\textwidth}{ \begin{center}\textbf{Hopf bifurcation and exch... ...joint work with Thomas Brand, Markus Kunze, Thorsten Seelbach) } \end{center}}$
We consider solutions bifurcating from a spatially homogeneous equilibrium under the assumption that the associated linearization possesses continuous spectrum up to the imaginary axis, for all values of the bifurcation parameter, and that a pair of imaginary eigenvalues crosses the imaginary axis. For a reaction-diffusion-convection system we investigate the nonlinear stability of the trivial solution with respect to spatially localized perturbations, prove the occurrence of a Hopf bifurcation and the nonlinear stability of the bifurcating time-periodic solutions, again with respect to spatially localized perturbations.

$\parbox{\textwidth}{ \begin{center}\textbf{The small viscosity limit... ...nelse{\equal{}{}}{} { \\ [\smallskipamount] (joint work with ) } \end{center}}$
Starting with a curved multiD shock solution to a hyperbolic system of conservation laws, we show how to obtain it as a limit as viscosity goes to zero of smooth solutions to an associated parabolic (hyperbolic + viscosity) problem. We first construct the viscous boundary layer on each side of the shock to high order (i.e., an approximate solution). The leading term in the boundary layer expansion is given by a family of travelling waves connecting endstates given by the original shock, and the problem reduces to proving nonlinear stability of this family. This is accomplished by proving L² estimates for the parabolic problem linearized about the approximate solution. A key step is to remove $x/\epsilon$ dependence from the coefficients of this problem by an argument that combines the Gap Lemma with pseudodifferential operators. One is then in a position to prove linear estimates using Kreiss-type symmetrizers, and these estimates suffice to yield nonlinear stability.

List of participants

THOMAS BRIDGES, Department of Mathematics and Statistics, School of ECM, University of Surrey, Guildford, Surrey, GU2 5XH, UK, t.bridges@surrey.ac.uk

ARJEN DOELMAN, Korteweg-de Vries Institute, Faculty of Sciences, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, the Netherlands,
doelman@science.uva.nl

SHIN-ICHIRO EI, Graduate School of Integrated Science, Yokohama City University, 22-2 Seto Kanazawa-ku, Yokohama 236-0027, Japan, ei2s@yokohama-cu.ac.jp

THIERRY GALLAY, Institut Fourier, Universit�é de Grenoble I, BP 74, 38402 Saint-Martin d'Hères, France, Thierry.Gallay@ujf-grenoble.fr

JOOST HULSHOF, Department of Mathematics, Faculty of Sciences, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands, jhulshof@cs.vu.nl

CHRISTOPHER JONES, Mathematics Department, University of North Carolina, Chapel Hill, NC 27599, USA, ckrtj@amath.unc.edu

TASSO KAPER, Department of Mathematics and Center for BioDynamics, Boston University, 111 Cummington Street, Boston, MA 02215, U.S.A. tasso@math.bu.edu

SHUNSAKU NII, Department of Mathematics, Faculty of Science, Saitama University, 255 Shimo-Okubo, Saitama 338-8570, Japan, snii@rimath.saitama-u.ac.jp

YASUMASA NISHIURA, Laboratory of Nonlinear Studies and Computation, Research Institute for Electronic Science, Hokkaido University, N12 W6 Sapporo, 060-0812, JAPAN, nishiura@aurora.es.hokudai.ac.jp

KEITH PROMISLOW, Simon Fraser University, Dept. of Mathematics, Burnaby BC, V5A 1S1, Canada, kpromisl@cs.sfu.ca

JEAN-MICHEL ROQUEJOFFRE, Laboratoire MIP, UFR MIG, Universite Paul Sabatier 118, route de Narbonne, 31062 Toulouse, Cedex, France, roque@mip.ups-tlse.fr

FREDERIC ROUSSET, Laboratoire J. A. Dieudonne, U.M.R. C.N.R.S. N 6621, E. D. P. Analyse Numerique, Universite de Nice Sophia-Antipolis, France, frousset@math.unice.fr

BJÖRN SANDSTEDE, Department of Mathematics, Ohio State University, 231 West 18th Avenue, Columbus, OH 43210, USA, sandsted@math.ohio-state.edu

ARND SCHEEL, U Minnesota, School of Mathematics, 206 Church St. S.E., Minneapolis, MN 55455, USA, scheel@math.umn.edu

GUIDO SCHNEIDER, Universtät Karlsruhe (TH), Mathematisches Institut I, D-76128 Karlsruhe, Germany, guido.schneider@mathematik.uni-karlsruhe.de

MARK WILLIAMS, Department of Mathematics, CB 3250, University of North Carolina, Chapel Hill, NC 27599, USA, williams@math.unc.edu