Whitham averaged equations and modulational stability of periodic traveling waves of a hyperbolic-parabolic balance law

Blake Barker[1]; Mathew A. Johnson[1]; Pascal Noble[2]; L.Miguel Rodrigues[3]; Kevin Zumbrun[1]

  • [1] Indiana University, Bloomington, IN 47405
  • [2] Université Lyon I, Villeurbanne, France
  • [3] Université de Lyon, Université Lyon 1, Institut Camille Jordan, UMR CNRS 5208, 43 bd du 11 novembre 1918, F - 69622 Villeurbanne Cedex, France

Journées Équations aux dérivées partielles (2010)

  • page 1-24
  • ISSN: 0752-0360

Abstract

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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.

How to cite

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Barker, Blake, et al. "Whitham averaged equations and modulational stability of periodic traveling waves of a hyperbolic-parabolic balance law." Journées Équations aux dérivées partielles (2010): 1-24. <http://eudml.org/doc/116384>.

@article{Barker2010,
abstract = {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.},
affiliation = {Indiana University, Bloomington, IN 47405; Indiana University, Bloomington, IN 47405; Université Lyon I, Villeurbanne, France; Université de Lyon, Université Lyon 1, Institut Camille Jordan, UMR CNRS 5208, 43 bd du 11 novembre 1918, F - 69622 Villeurbanne Cedex, France; Indiana University, Bloomington, IN 47405},
author = {Barker, Blake, Johnson, Mathew A., Noble, Pascal, Rodrigues, L.Miguel, Zumbrun, Kevin},
journal = {Journées Équations aux dérivées partielles},
keywords = {Periodic traveling waves; St. Venant equations; Spectral stability; Nonlinear stability},
language = {eng},
month = {6},
pages = {1-24},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Whitham averaged equations and modulational stability of periodic traveling waves of a hyperbolic-parabolic balance law},
url = {http://eudml.org/doc/116384},
year = {2010},
}

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
DA - 2010/6//
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 24
AB - 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.
LA - eng
KW - Periodic traveling waves; St. Venant equations; Spectral stability; Nonlinear stability
UR - http://eudml.org/doc/116384
ER -

References

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