Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation

Abstract

In a companion paper, we established nonlinear stability with detailed diffusive rates of decay of spectrally stable periodic traveling-wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized perturbation. Here, we determine time-asymptotic behavior under such perturbations, showing that solutions consist to leading order of a modulation whose parameter evolution is governed by an associated Whitham averaged equation.

References in corpus (4)

• Nonlocalized modulation of periodic reaction diffusion waves: Nonlinear stability
• Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations
• Stability of Periodic Kuramoto-Sivashinsky Waves
• Whitham's equations for modulated roll-waves in shallow flows

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• Modulated equations of Hamiltonian PDEs and dispersive shocks
• Diffusive stability of spatially periodic solutions of the Brusselator model
• Subharmonic Dynamics of Wave Trains in Reaction Diffusion Systems
• Toward nonlinear stability of sources via a modified Burgers equation
• Diffusive stability against nonlocalized perturbations of planar wave trains in reaction-diffusion systems
• Nonlinear stability and asymptotic behavior of periodic wave trains in reaction-diffusion systems against $C_{\mathrm{ub}}$-perturbations
• Nonlinear Subharmonic Dynamics of Spectrally Stable Lugiato-Lefever Periodic Waves
• Diffusive stability of spatially periodic patterns with a conservation law
• Nonlinear stability of periodic roll solutions in the real Ginzburg-Landau equation against $C_{\mathrm{ub}}^m$-perturbations
• Nonlinear stability of source defects in oscillatory media
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• Stability of Small Periodic Waves in Fractional KdV Type Equations
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