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Contact defects are time-periodic patterns in one space dimension that resemble spatially homogeneous oscillations with a defect embedded in their core region. For theoretical and numerical purposes, it is important to understand whether these defects persist when the domain is truncated to large spatial intervals, supplemented by appropriate boundary conditions. The present work shows that truncated contact defects exist and are unique on sufficiently large spatial intervals. [ABSTRACT FROM AUTHOR]

Spiral wave patterns observed in models of cardiac arrhythmias and chemical oscillations develop alternans and stationary line defects, which can both be thought of as period-doubling instabilities. These instabilities are observed on bounded domains and may be caused by the spiral core, far-field asymptotics, or boundary conditions. Here, we introduce a methodology to disentangle the impacts of each region on the instabilities by analyzing spectral properties of spiral waves and boundary sinks on bounded domains with appropriate boundary conditions. We apply our techniques to spirals formed in reaction-diffusion systems to investigate how and why alternans and line defects develop. Our results indicate that the mechanisms driving these instabilities are quite different; alternans are driven by the spiral core, whereas line defects appear from boundary effects. Moreover, we find that the shape of the alternans eigenfunction is due to the interaction of a point eigenvalue with curves of continuous spectra. [ABSTRACT FROM AUTHOR]

The FitzHugh{Nagumo equations are known to admit fast traveling pulse solutions with monotone tails. It is also known that this system admits traveling pulses with exponentially decaying oscillatory tails. Upon numerical continuation in parameter space, it has been observed that the oscillations in the tails of the pulses grow into a secondary excursion resembling a second copy of the primary pulse. In this paper, we outline in detail the geometric mechanism responsible for this single-to-double-pulse transition, and we construct the transition analytically using geometric singular perturbation theory and blow-up techniques. [ABSTRACT FROM AUTHOR]

Numerical studies indicate that the FitzHugh-Nagumo system exhibits stable traveling pulses with oscillatory tails. In this paper, the existence of such pulses is proved analytically in the singular perturbation limit near parameter values where the FitzHugh-Nagumo system exhibits folds. In addition, the stability of these pulses is investigated numerically, and a mechanism is proposed that explains the transition from single to double pulses that was observed in earlier numerical studies. The existence proof utilizes geometric blow-up techniques combined with the exchange lemma: the main challenge is to understand the passage near two fold points on the slow manifold where normal hyperbolicity fails. [ABSTRACT FROM AUTHOR]

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 patterns which are periodic in the transverse direction and use it to calculate the Maxwell curves along which the selected hexagons have the same energy as the trivial state. We find that the Maxwell curve lies within the snaking region, as expected from heuristic arguments. [ABSTRACT FROM AUTHOR]

Motivated by experimental observations in the light-sensitive Belousov—Zhabotinsky reaction and subsequent numerical works, we discuss period-doubling bifurcations of spiral waves and other coherent structures. We report on explanations of the observed phenomena which involve a detailed analysis of spectra, and of the associated eigenfunctions, of defects on bounded and unbounded domains. [ABSTRACT FROM AUTHOR]

Fronts are traveling waves in spatially extended systems that connect two different spatially homogeneous rest states. If the rest state behind the front undergoes a supercritical Turing instability, then the front will also destabilize. On the linear level, however, the front will be only convectively unstable since perturbations will be pushed away from the front as it propagates. In other words, perturbations may grow, but they can do so only behind the front. The goal of this paper is to prove for a specific model system that this behavior carries over to the full nonlinear system. [ABSTRACT FROM AUTHOR]

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 cubic- quintic Ginzburg-Landau equation as a perturbation of the phase-slip vortex. [ABSTRACT FROM AUTHOR]

In this article, existence and stability of N-front travelling-wave solutions of partial differential equations on the real line is investigated. The N-fronts considered here arise as heteroclinic orbits bifurcating from a twisted heteroclinic loop in the underlying ordinary differential equation describing travelling-wave solutions. It is proved that the N-front solutions are linearly stable provided the fronts building the twisted heteroclinic loop are linearly stable. The result is applied to travelling waves arising in the FitzHugh-Nagumo equation. [ABSTRACT FROM AUTHOR]

Motivated by pulse-replication phenomena observed in the FitzHugh--Nagumo equation, we investigate traveling pulses whose slow/fast profiles exhibit canard-like transitions. We show that the spectra of the PDE linearization about such pulses may contain many point eigenvalues that accumulate onto a union of curves as the slow scale parameter approaches zero. The limit sets are related to the absolute spectrum of the homogeneous rest states involved in the canard-like transitions. Our results are formulated for general systems that admit an appropriate slow/fast structure. [ABSTRACT FROM AUTHOR]

In its original formulation the Krein matrix was used to locate the spectrum of first-order star-even polynomial operators where both operator coefficients are nonsingular. Such operators naturally arise when considering first-order-in-time Hamiltonian PDEs. Herein the matrix is reformulated to allow for operator coefficients with nontrivi al kernel. Moreover, it is extended to allow for the study of the spectral problem associated with quadratic star-even operators, which arise when considering the spectral problem associated with second-order-in-time Hamiltonian PDEs. In conjunction with the Hamiltonian-Krein index (HKI) the Krein matrix is used to study two problems: conditions leading to Hamiltonian-Hopf bifurcations for small spatially periodic waves, and the location and Krein signature of small eigenvalues associated with, e.g., n-pulse problems. For the first case we consider in detail a first-order-in-time fifth-order KdV-like equation. In the latter case we use a combination of Lin's method, the HKI, and the Krein matrix to study the spectrum associated with n-pulses for a second-order-in-time Hamiltonian system which is used to model the dynamics of a suspension bridge. [ABSTRACT FROM AUTHOR]

The nonlinear stability of traveling Lax shocks in semidiscrete conservation laws involving general spatial forward-backward discretization schemes is considered. It is shown that spectrally stable semidiscrete Lax shocks are nonlinearly stable. In addition, it is proved that weak semidiscrete Lax profiles satisfy the spectral stability hypotheses made here and are therefore non-linearly stable. The nonlinear stability results are proved by constructing the resolvent kernel using exponential dichotomies, which have recently been developed in this setting, and then using the contour integral representation for the associated Green's function to derive pointwise bounds that are su?cient for proving nonlinear stability. Previous stability analyses for semidiscrete shocks relied primarily on Evans functions, which exist only for one-sided upwind schemes. [ABSTRACT FROM AUTHOR]

The numerical computation of solitary waves to semilinear elliptic equations in infinite cylindrical domains is investigated. Rather than solving on the infinite cylinder, the equation is approximated by a boundary-value problem on a finite cylinder. Convergence and stability results for this approach are given. It is also shown that Galerkin approximations can be used to compute solitary waves of the elliptic problem on the finite cylinder. In addition, it is demonstrated that the aforementioned procedures simplify in cases where the elliptic equation admits an additional reversibility structure. Finally, the theoretical predictions are compared with numerical computations. In particular, post buckling of an infinitely long cylindrical shell under axial compression is considered; it is shown numerically that, for a fixed spatial truncation, the error in the truncation scales with the length of the cylinder as predicted theoretically. [ABSTRACT FROM AUTHOR]

The optimal-velocity follow-the-leader model is augmented with an equation that allows each driver to adjust their target headway according to the velocity difference between the driver and the car in front. In this more detailed model, which is investigated on a ring, stable and unstable multipulse or multijam solutions emerge. Analytical investigations using truncated Fourier analysis are confirmed and complemented by a detailed numerical bifurcation analysis. In addition to standard rotating waves, time-modulated waves are found. [ABSTRACT FROM AUTHOR]

Stable localized roll structures have been observed in many physical problems and model equations, notably in the one-dimensional (1D) Swift--Hohenberg equation. Reflection-symmetric localized rolls are often found to lie on two "snaking" solution branches so that the spatial width of the localized rolls increases when moving along each branch. Recent numerical results by Burke and Knobloch indicate that the two branches are connected by infinitely many "ladder" branches of asymmetric localized rolls. In this paper, these phenomena are investigated analytically. It is shown that both snaking of symmetric pulses and the ladder structure of asymmetric states can be predicted completely from the bifurcation structure of fronts that connect the trivial state to rolls. It is also shown that isolas of asymmetric states may exist, and it is argued that the results presented here apply to 2D stationary states that are localized in one spatial direction. [ABSTRACT FROM AUTHOR]