9,660 research outputs found
Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations
Using spatial domain techniques developed by the authors and Myunghyun Oh in
the context of parabolic conservation laws, we establish under a natural set of
spectral stability conditions nonlinear asymptotic stability with decay at
Gaussian rate of spatially periodic traveling-waves of systems of reaction
diffusion equations. In the case that wave-speed is identically zero for all
periodic solutions, we recover and slightly sharpen a well-known result of
Schneider obtained by renormalization/Bloch transform techniques; by the same
arguments, we are able to treat the open case of nonzero wave-speeds to which
Schneider's renormalization techniques do not appear to appl
Nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case
Extending previous results of Oh--Zumbrun and Johnson--Zumbrun, we show that
spectral stability implies linearized and nonlinear stability of spatially
periodic traveling-wave solutions of viscous systems of conservation laws for
systems of generic type, removing a restrictive assumption that wave speed be
constant to first order along the manifold of nearby periodic solutions.Comment: Fixed minor typo
Whitham Averaged Equations and Modulational Stability of Periodic Traveling Waves of a Hyperbolic-Parabolic Balance Law
In this note, we report on recent findings concerning the spectral and
nonlinear stability of periodic traveling wave solutions of
hyperbolic-parabolic systems of balance laws, as applied to the St. Venant
equations of shallow water flow down an incline. We begin by introducing a
natural set of spectral stability assumptions, motivated by considerations from
the Whitham averaged equations, and outline the recent proof yielding nonlinear
stability under these conditions. We then turn to an analytical and numerical
investigation of the verification of these spectral stability assumptions.
While spectral instability is shown analytically to hold in both the Hopf and
homoclinic limits, our numerical studies indicates spectrally stable periodic
solutions of intermediate period. A mechanism for this moderate-amplitude
stabilization is proposed in terms of numerically observed "metastability" of
the the limiting homoclinic orbits.Comment: 27 pages, 5 figures. Minor changes throughou
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