Nonlocalized modulation of periodic reaction diffusion waves: Nonlinear stability

Abstract

By a refinement of the technique used by Johnson and Zumbrun to show stability under localized perturbations, we show that spectral stability implies nonlinear modulational stability of periodic traveling-wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized perturbation. The main new ingredient is a detailed analysis of linear behavior under modulational data ${\bar{u}}^{\prime }(x)h_0(x)$, where $\bar{u}$ is the background profile and $h_0$ is the initial modulation

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• Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation
• Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations
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Cited by in corpus (14)

• Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation
• Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation
• Modulational Instability in Equations of KdV Type
• Nonlinear stability of source defects in the complex Ginzburg-Landau equation
• Diffusive stability of spatially periodic solutions of the Brusselator model
• Spectral instability of small-amplitude periodic waves of the electronic Euler-Poisson system
• 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
• Turing patterns in parabolic systems of conservation laws and numerically observed stability of periodic waves
• 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
• Nonlinear stability of source defects in oscillatory media
• Diffusive mixing of periodic wave trains in reaction-diffusion systems