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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2402.07567 |
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| _version_ | 1866914674970198016 |
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| author | Li, Huaiyu Hofstrand, Andrew Weinstein, Michael I. |
| author_facet | Li, Huaiyu Hofstrand, Andrew Weinstein, Michael I. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07567 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stability of traveling waves in a nonlinear hyperbolic system approximating a dimer array of oscillators Li, Huaiyu Hofstrand, Andrew Weinstein, Michael I. Pattern Formation and Solitons Mathematical Physics Analysis of PDEs Spectral Theory 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. |
| title | Stability of traveling waves in a nonlinear hyperbolic system approximating a dimer array of oscillators |
| topic | Pattern Formation and Solitons Mathematical Physics Analysis of PDEs Spectral Theory |
| url | https://arxiv.org/abs/2402.07567 |