Communications in Mathematical Sciences

Volume 10 (2012)

Number 3

Shock dynamics in layered periodic media

Pages: 859 – 874



David I. Ketcheson (Mathematical and Computer Sciences Division, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia)

Randall J. Leveque (Department of Applied Mathematics, University of Washington, Seattle, Washington)


Solutions of constant-coeffcient nonlinear hyperbolic PDEs generically develop shocks, even if the initial data is smooth. Solutions of hyperbolic PDEs with variable coeffcients can behave very differently. We investigate formation and stability of shock waves in a one-dimensional periodic layered medium by a computational study of time-reversibility and entropy evolution. We find that periodic layered media tend to inhibit shock formation. For small initial conditions and large impedance variation, no shock formation is detected even after times much greater than the time of shock formation in a homogeneous medium. Furthermore, weak shocks are observed to be dynamically unstable in the sense that they do not lead to significant long-term entropy decay. We propose a characteristic condition for admissibility of shocks in heterogeneous media that generalizes the classical Lax entropy condition and accurately predicts the formation or absence of shocks in these media.


shock waves, periodic media, dispersive shocks, solitary waves

2010 Mathematics Subject Classification

35L02, 35L67, 37K40, 65M08, 74E15

Published 9 April 2012