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

Author:

Huaiyu Li, Andrew Hofstrand, Michael I. Weinstein

Keyword:

Nonlinear Sciences, Pattern Formation and Solitons, Pattern Formation and Solitons (nlin.PS), Mathematical Physics (math-ph), Analysis of PDEs (math.AP), Spectral Theory (math.SP)

journal:

date:

2024-02-12 00:00:00

Abstract

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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