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<p>We study a semilinear hyperbolic system of PDEs which arises as a continuum approximation of the discrete nonlinear dimer array model introduced by Hadad et al. (ACS Photonics 4:1974–1979, 2017). 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 equilibria 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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="332_2025_10150_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\(H^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> norms of the perturbation, which capture the phenomenon of <i>convective stabilization</i>; as time advances, the traveling wave “outruns” the <Emphasis Type="Underline">growing</Emphasis> disturbance excited by an initial perturbation; the nontrivial spatially uniform equilibria are linearly exponentially unstable. We use our analytical results to interpret phenomena observed in numerical simulations.</p>

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Stability of Traveling Waves in a Nonlinear Hyperbolic System Approximating a Dimer Array of Oscillators

  • 期刊论文

  • 发表日期：

    2025年4月3日

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  • Huaiyu Li,
  • Andrew Hofstrand,
  • Michael I. Weinstein

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 et al. (ACS Photonics 4:1974–1979, 2017). 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 equilibria 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 \(H^1\) H 1 norms of the perturbation, which capture the phenomenon of convective stabilization; as time advances, the traveling wave “outruns” the growing disturbance excited by an initial perturbation; the nontrivial spatially uniform equilibria are linearly exponentially unstable. We use our analytical results to interpret phenomena observed in numerical simulations.