非线性双曲系统中逼近振子二聚体阵列的行波稳定性
Stability of traveling waves in a nonlinear hyperbolic system approximating a dimer array of oscillators
AI总结：
研究逼近离散非线性二聚体阵列的半线性双曲型偏微分方程组，分类并分析了行波脉冲与移动畴壁的谱稳定性及非线性稳定性，揭示了对流稳定化现象。
摘要：
我们研究了一个半线性双曲型偏微分方程组，它是 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逼近。我们对系统的行波进行了分类，并研究了它们的稳定性性质。我们重点关注非平凡背景上的行波脉冲解（“孤子”）和移动畴壁解（扭结）；两者都作为约化动力学系统的空间均匀平衡点之间的异宿连接出现。我们给出了关于以下方面的解析结果：超音速脉冲的非线性稳定性和谱稳定性，以及移动畴壁的谱稳定性。我们的稳定性结果用扰动的加权 $H^1$ 范数表示，这捕捉了“对流稳定化”现象；随着时间推移，行波“跑赢”了由初始扰动激发的不断增长的扰动；非平凡的空间均匀平衡点是线性指数不稳定的。我们利用我们的解析结果来解释在数值模拟中观察到的现象。
英文摘要：
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.
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