SIAM Journal on Applied Mathematics, Volume 80, Issue 4, Page 1704-1722, January 2020.
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.

中文翻译：

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亥姆霍兹共振器阵列中非线性声学的Sugimoto模型分析

SIAM应用数学杂志，第80卷，第4期，第1704-1722页，2020年1月。
理论上研究了包含非线性标量PDE和线性ODE的耦合系统。这个具有松弛的双曲系统模拟了非线性波在连接到亥姆霍兹谐振器的波导中的传播，该设备是非线性声学超材料的一个例子。在上一篇论文中[N. Sugimoto，J。Fluid。Mech。，244（1992），pp。55--78]，已表明该装置还允许声孤子的传播。在本文中，分析了耦合系统的数学性质：在有限时间内形成奇异性，分数界变分空间中熵解的存在以及单个熵族的唯一性。关于弱耦合系统，也得出了新的结果。数值模拟说明了这些发现。