标题： 非线性双曲系统中行波的稳定性近似二聚体振子阵列 显示英文标题

标题： Stability of traveling waves in a nonlinear hyperbolic system approximating a dimer array of oscillators

摘要： 我们研究了一个半线性双曲型偏微分方程组，该方程组作为Hadad、Vitelli和Alu（HVA）在\cite{HVA17}中引入的离散非线性二聚体阵列模型的连续近似而出现。我们对系统的行波进行分类，并研究它们的稳定性特性。我们关注在非平凡背景上移动的行波脉冲解（“孤子”）和移动域壁解（kinks）；这两种解都作为约化动力系统空间均匀平衡之间的异宿连接出现。我们提出了关于：超音速脉冲的非线性稳定性和谱稳定性，以及移动域壁的谱稳定性的分析结果。我们的稳定性结果是基于扰动的加权$H^1$范数，这些范数捕捉了{\it 对流稳定化}现象；随着时间的推移，行波“超越”了由初始扰动激发的\underline{增长}扰动；非平凡的空间均匀平衡线性指数不稳定。我们利用分析结果来解释数值模拟中观察到的现象。 显示英文摘要

摘要： We study a semilinear hyperbolic system of PDEs which arises as a continuum approximation of the discrete nonlinear dimer array model introduced by Hadad, Vitelli and Alu (HVA) in \cite{HVA17}. We classify the system's traveling waves, and study their stability properties. We focus on traveling pulse solutions (``solitons'') on a nontrivial background and moving domain wall solutions (kinks); both arise as heteroclinic connections between spatially uniform equilibrium of a reduced dynamical system. We present analytical results on: nonlinear stability and spectral stability of supersonic pulses, and spectral stability of moving domain walls. Our stability results are in terms of weighted $H^1$ norms of the perturbation, which capture the phenomenon of {\it convective stabilization}; as time advances, the traveling wave ``outruns'' the \underline{growing} disturbance excited by an initial perturbation; the non-trivial spatially uniform equilibria are linearly exponentially unstable. We use our analytical results to interpret phenomena observed in numerical simulations.