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. ∗Indiana University, Bloomington, IN 47405; [email protected]: Research of B.B. was partially supported under NSF grants no. DMS-0300487 and DMS-0801745. †Indiana University, Bloomington, IN 47405; [email protected]: Research of M.J. was partially supported by an NSF Postdoctoral Fellowship under NSF grant DMS-0902192. ‡Université Lyon I, Villeurbanne, France; [email protected]: Research of P.N. was partially supported by the French ANR Project no. ANR-09-JCJC-0103-01. §Université de Lyon, Université Lyon 1, Institut Camille Jordan, UMR CNRS 5208, 43 bd du 11 novembre 1918, F 69622 Villeurbanne Cedex, France; [email protected] ¶Indiana University, Bloomington, IN 47405; [email protected]: Research of K.Z. was partially supported under NSF grants no. DMS-0300487 and DMS-0801745.