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Analysis of a Sugimoto's model of nonlinear acoustics in an array of Helmholtz resonators

Abstract : A coupled system involving a nonlinear scalar PDE and a linear ODE is theoretically investigated. This hypebolic 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 [Sugimoto, J. Fluid. Mech. 1992], it has been shown that this device allows also the propagation of acoustic solitons. In the present paper, the mathematical properties of the coupled system are analysed: formation of singularity in finite time, existence of global smooth solutions for small data, existence of entropy solutions in fractional BV spaces and uniqueness with a single family of entropies. New results are also deduced about weakly coupled systems. Numerical simulations illustrate these findings.
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Submitted on : Tuesday, April 21, 2020 - 7:27:18 AM
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Stéphane Junca, Bruno Lombard. Analysis of a Sugimoto's model of nonlinear acoustics in an array of Helmholtz resonators. SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2020, 80 (4), pp.1704-1722. ⟨10.1137/19M1280624⟩. ⟨hal-02186692v2⟩

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