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Planar wave trains are traveling wave solutions whose wave profiles are periodic in one spatial direction and constant in the transverse direction. In this paper, we investigate the stability of planar wave trains in reaction-diffusion systems. We establish nonlinear diffusive stability against perturbations that are bounded along a line in R 2 and decay exponentially in the distance from this line. Our analysis is the first to treat spatially nonlocalized perturbations that do not originate from a phase modulation. We also consider perturbations that are fully localized and establish nonlinear stability with better decay rates, suggesting a trade-off between spatial localization of perturbations and temporal decay rate. Our stability analysis utilizes pointwise estimates to exploit the spatial structure of the perturbations. The nonlocalization of perturbations prevents the use of damping estimates in the nonlinear iteration scheme; instead, we track the perturbed solution in two different coordinate systems. [ABSTRACT FROM AUTHOR]

Planar wave trains are traveling wave solutions whose wave profiles are periodic in one spatial direction and constant in the transverse direction. In this paper, we investigate the stability of planar wave trains in reaction–diffusion systems. We establish nonlinear diffusive stability against perturbations that are bounded along a line in R 2 and decay exponentially in the distance from this line. Our analysis is the first to treat spatially nonlocalized perturbations that do not originate from a phase modulation. We also consider perturbations that are fully localized and establish nonlinear stability with better decay rates, suggesting a trade-off between spatial localization of perturbations and temporal decay rate. Our stability analysis utilizes pointwise estimates to exploit the spatial structure of the perturbations. The nonlocalization of perturbations prevents the use of damping estimates in the nonlinear iteration scheme; instead, we track the perturbed solution in two different coordinate systems. [ABSTRACT FROM AUTHOR]

Modeling networks of synaptically coupled neurons often leads to systems of integro-differential equations. Particularly interesting solutions in this context are traveling waves. We prove here that spectral stability of traveling waves implies their nonlinear stability in appropriate function spaces, and compare several recent Evans-function constructions that are useful tools when analyzing spectral stability. [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]

Abstract: Coherent structures are solutions to reaction–diffusion systems that are time-periodic in an appropriate moving frame and spatially asymptotic at to spatially periodic travelling waves. This paper is concerned with sources which are coherent structures for which the group velocities in the far field point away from the core. Sources actively select wave numbers and therefore often organize the overall dynamics in a spatially extended system. Determining their nonlinear stability properties is challenging as localized perturbations may lead to a non-localized response even on the linear level due to the outward transport. Using a Burgers-type equation as a model problem that captures some of the essential features of sources, we show how this phenomenon can be analysed and asymptotic nonlinear stability be established in this simpler context. [Copyright &y& Elsevier]

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]