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.
Mots-clés : Periodic traveling waves, St. Venant equations, Spectral stability, Nonlinear stability.
@incollection{JEDP_2010____A3_0,
author = {Barker, Blake and Johnson, Mathew A. and Noble, Pascal and Rodrigues, L.Miguel and Zumbrun, Kevin},
title = {Whitham averaged equations and modulational stability of periodic traveling waves of a hyperbolic-parabolic balance law},
booktitle = {},
series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
eid = {3},
pages = {1--24},
publisher = {Groupement de recherche 2434 du CNRS},
year = {2010},
doi = {10.5802/jedp.60},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jedp.60/}
}
TY - JOUR AU - Barker, Blake AU - Johnson, Mathew A. AU - Noble, Pascal AU - Rodrigues, L.Miguel AU - Zumbrun, Kevin TI - Whitham averaged equations and modulational stability of periodic traveling waves of a hyperbolic-parabolic balance law JO - Journées équations aux dérivées partielles PY - 2010 SP - 1 EP - 24 PB - Groupement de recherche 2434 du CNRS UR - http://www.numdam.org/articles/10.5802/jedp.60/ DO - 10.5802/jedp.60 LA - en ID - JEDP_2010____A3_0 ER -
%0 Journal Article %A Barker, Blake %A Johnson, Mathew A. %A Noble, Pascal %A Rodrigues, L.Miguel %A Zumbrun, Kevin %T Whitham averaged equations and modulational stability of periodic traveling waves of a hyperbolic-parabolic balance law %J Journées équations aux dérivées partielles %D 2010 %P 1-24 %I Groupement de recherche 2434 du CNRS %U http://www.numdam.org/articles/10.5802/jedp.60/ %R 10.5802/jedp.60 %G en %F JEDP_2010____A3_0
Barker, Blake; Johnson, Mathew A.; Noble, Pascal; Rodrigues, L.Miguel; Zumbrun, Kevin. Whitham averaged equations and modulational stability of periodic traveling waves of a hyperbolic-parabolic balance law. Journées équations aux dérivées partielles (2010), article no. 3, 24 p. doi : 10.5802/jedp.60. http://www.numdam.org/articles/10.5802/jedp.60/
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