Björn Sandstede

Chapter 18 - Stability of Travelling Waves

Abstract An overview of various aspects related to the spectral and nonlinear stability of travelling-wave solutions to partial differential equations is given. The point and the essential spectrum of the linearization about a travelling wave are discussed as is the relation between these spectra, Fredholm properties, and the existence of exponential dichotomies (or Greens functions) for the linear operator. Among the other topics reviewed in this survey are the nonlinear stability of waves, the stability and interaction of well-separated multi-bump pulses, the numerical computation of spectra, and the Evans function, which is a tool to locate isolated eigenvalues in the point spectrum and near the essential spectrum. Furthermore, methods for the stability of waves in Hamiltonian and monotone equations as well as for singularly perturbed problems are mentioned. Modulated waves, rotating waves on the plane, and travelling waves on cylindrical domains are also discussed briefly.

Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations

Abstract The propagation of pulses in ideal nonlinear optical fibers without loss is governed by the nonlinear Schrodinger equation (NLS). When considering realistic fibers one must examine perturbed NLS equations, with the particular perturbation depending on the physical situation that is being modeled. A common example is the complex Ginzburg-Landau equation (CGL), which is a dissipative perturbation. It is known that some of the stable bright solitons of the NLS survive a dissipative perturbation such as the CGL. Given that a wave persists, it is then important to determine its stability with respect to the perturbed NLS. A major difficulty in analyzing the stability of solitary waves upon adding dissipative terms is that eigenvalues may bifurcate out of the essential spectrum. Since the essential spectrum of the NLS is located on the imaginary axis, such eigenvalues may lead to an unstable wave. In fact, eigenvalues can pop out of the essential spectrum even if the unperturbed problem has no eigenvalue embedded in the essential spectrum. Here we present a technique which can be used to track these bifurcating eigenvalues. As a consequence, we are able to locate the spectrum for bright solitary-wave solutions to various perturbed nonlinear Schrodinger equations, and determine precise conditions on parameters for which the waves are stable. In addition, we show that a particular perturbation, the parametrically forced NLS equation, supports stable multi-bump solitary waves. The technique for tracking eigenvalues which bifurcate from the essential spectrum is very general and should therefore be applicable to a larger class of problems than those presented here.

Absolute and convective instabilities of waves on unbounded and large bounded domains

Abstract Instabilities of nonlinear waves on unbounded domains manifest themselves in different ways. An absolute instability occurs if the amplitude of localized wave packets grows in time at each fixed point in the domain. In contrast, convective instabilities are characterized by the fact that even though the overall norm of wave packets grows in time, perturbations decay locally at each given point in the unbounded domain: wave packets are convected towards infinity. In experiments as well as in numerical simulations, bounded domains are often more relevant. We are interested in the effects that the truncation of the unbounded to a large but bounded domain has on the aforementioned (in)stability properties of a wave. These effects depend upon the boundary conditions that are imposed on the bounded domain. We compare the spectra of the linearized evolution operators on unbounded and bounded domains for two classes of boundary conditions. It is proved that periodic boundary conditions reproduce the point and essential spectrum on the unbounded domain accurately. Spectra for separated boundary conditions behave in quite a different way: firstly, separated boundary conditions may generate additional isolated eigenvalues. Secondly, the essential spectrum on the unbounded domain is in general not approximated by the spectrum on the bounded domain. Instead, the so-called absolute spectrum is approximated that corresponds to the essential spectrum on the unbounded domain calculated with certain optimally chosen exponential weights. We interpret the difference between the absolute and the essential spectrum in terms of the convective behavior of the wave on the unbounded domain. In particular, it is demonstrated that the stability of the absolute spectrum implies convective instability of the wave, but that convectively unstable waves can destabilize under domain truncation. The theoretical predictions are compared with numerical computations.

A numerical toolbox for homoclinic bifurcation analysis

This paper presents extensions and improvements of recently developed algorithms for the numerical analysis of orbits homoclinic to equilibria in ODEs and describes the implementation of these algorithms within the standard continuation package AUTO86. This leads to a kind of toolbox, called HOMCONT, for analysing homoclinic bifurcations either as an aid to producing new theoretical results, or to understand dynamics arising from applications. This toolbox allows the continuation of codimension-one homoclinic orbits to hyperbolic or non-hyperbolic equilibria as well as detection and continuation of higher-order homoclinic singularities in more parameters. All known codimension-two cases that involve a unique homoclinic orbit are supported. Two specific example systems from ecology and chemical kinetics are analysed in some detail, allowing the reader to understand how to use the the toolbox for themselves. In the process, new results are also derived on these two particular models.

Chapter 4 – Numerical Continuation, and Computation of Normal Forms

This chapter describes numerical continuation methods for analyzing the solution behavior of the dynamical system. Time-integration of a dynamical system gives much insight into its solution behavior. However, once a solution type has been computed—for example, a stationary solution (equilibrium) or a periodic solution (cycle)—then continuation methods become very effective in determining the dependence of this solution on the parameter α. Once a co-dimension-1 bifurcation has been located, it can be followed in two parameters—that is, with α e ℝ 2 . However, in many cases, detection of higher co-dimension bifurcations requires computation of certain normal forms for equations restricted to center manifolds at the critical parameter values. Pseudo-arclength continuation method allows the continuation of any regular solution, including folds. Geometrically, it is the most natural continuation method. The periodic solution continuation method is very suitable for numerical computations, and it is not difficult to establish the Poincare continuation with the help of it.

Stability of multiple-pulse solutions

In this article, stability of multiple-pulse solutions in semilinear parabolic equations on the real line is studied. A system of equations is derived which determines stability of N-pulses bifurcating from a stable primary pulse. The system depends only on the particular bifurcation leading to the existence of the N-pulses. As an example, existence and stability of multiple pulses are investigated if the primary pulse converges to a saddle-focus. It turns out that under suitable assumptions infinitely many N-pulses bifurcate for any fixed N > 1. Among them are infinitely many stable ones. In fact, any number of eigenvalues between 0 and N − 1 in the right half plane can be prescribed.

Localized hexagon patterns of the planar Swift-Hohenberg equation

We investigate stationary spatially localized hexagon patterns of the two-dimensional (2D) Swift–Hohenberg equation in the parameter region where the trivial state and regular hexagon patterns are both stable. Using numerical continuation techniques, we trace out the existence regions of fully localized hexagon patches and of planar pulses which consist of a strip filled with hexagons that is embedded in the trivial state. We find that these patterns exhibit snaking: for each parameter value in the snaking region, an infinite number of patterns exist that are connected in parameter space and whose width increases without bound. Our computations also indicate a relation between the limits of the snaking regions of planar hexagon pulses with different orientations and of the fully localized hexagon patches. To investigate which hexagons among the one-parameter family of hexagons are selected in a hexagon pulse or front, we derive a conserved quantity of the spatial dynamical system that describes planar pat...

Homoclinic and heteroclinic bifurcations in vector fields

An overview of homoclinic and heteroclinic bifurcation theory for autonomous vector fields is given. Specifically, homoclinic and heteroclinic bifurcations of codimension one and two in generic, equivariant, reversible, and conservative systems are reviewed, and results pertaining to the existence of multi-round homoclinic and periodic orbits and of complicated dynamics such as suspended horseshoes and attractors are stated. Bifurcations of homoclinic orbits from equilibria in local bifurcations are also considered. The main analytic and geometric techniques such as Lin’s method, Shil’nikov variables and homoclinic centre manifolds for analyzing these bifurcations are discussed. Finally, a few related topics, such as topological moduli, numerical algorithms, variational methods, and extensions to singularly perturbed and infinite-dimensional systems, are reviewed briefly.

The dynamics of modulated wave trains.

The authors of this title investigate the dynamics of weakly-modulated nonlinear wave trains. For reaction-diffusion systems and for the complex Ginzburg - Landau equation, they establish rigorously that slowly varying modulations of wave trains are well approximated by solutions to the Burgers equation over the natural time scale. In addition to the validity of the Burgers equation, they show that the viscous shock profiles in the Burgers equation for the wave number can be found as genuine modulated waves in the underlying reaction-diffusion system. In other words, they 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, the authors also consider similar shock profiles for wave trains with large wave number, that is, for an infinite sequence of widely separated pulses. The results presented here are applied to the FitzHugh - Nagumo equation and to hydrodynamic stability problems.

Defects in Oscillatory Media: Toward a Classification ∗

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...