Subharmonic Dynamics of Wave Trains in Reaction Diffusion Systems

Abstract

We investigate the stability and nonlinear local dynamics of spectrally stable wave trains in reaction-diffusion systems. For each $N\in \mathbb{N}$, such $T$-periodic traveling waves are easily seen to be nonlinearly asymptotically stable (with asymptotic phase) with exponential rates of decay when subject to $NT$-periodic, i.e., subharmonic, perturbations. However, both the allowable size of perturbations and the exponential rates of decay depend on $N$, and, in particular, they tend to zero as $N\to \infty$, leading to a lack of uniformity in such subharmonic stability results. In this work, we build on recent work by the authors and introduce a methodology that allows us to achieve a stability result for subharmonic perturbations which is uniform in $N$. Our work is motivated by the dynamics of such waves when subject to perturbations which are localized (i.e. integrable on the line), which has recently received considerable attention by many authors.

arXiv comment24 pages. Updated exposition and incorporated referee comments

References in corpus (4)

• Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation
• Nonlocalized modulation of periodic reaction diffusion waves: Nonlinear stability
• Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations
• Diffusive stability of Turing patterns via normal forms

Cited by in corpus (2)

• Nonlinear Subharmonic Dynamics of Spectrally Stable Lugiato-Lefever Periodic Waves
• Convergence of linear solutions through convergence of periodic initial data