Type: Article
Publication Date: 2020-01-01
Citations: 5
DOI: https://doi.org/10.1137/19m1280624
A coupled system involving a nonlinear scalar PDE and a linear ODE is theoretically investigated. This hyperbolic system with relaxation models the propagation of nonlinear waves in a waveguide connected to Helmholtz resonators, this device being an example of a nonlinear acoustic metamaterial. In a previous paper [N. Sugimoto, J. Fluid. Mech., 244 (1992), pp. 55--78], it has been shown that this device also allows the propagation of acoustic solitons. In the present paper, the mathematical properties of the coupled system are analyzed: formation of singularity in finite time, existence of entropy solutions in fractional bounded variation spaces, and uniqueness with a single family of entropies. New results are also deduced about weakly coupled systems. Numerical simulations illustrate these findings.