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

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.

Figures (13)

• Figure 1: Convective stability of supersonic pulses. Snapshots of a perturbed supersonic ($c=2>c_0=1$) pulse of (\ref{['eq: PDE in lab frame']}) in a reference frame of speed $c$. Perturbation at $t=0$ is concentrated to the right of the core. Red curves indicate the amplitude of the solution at different times. Blue dashed curves indicate the amplitude profile of the unperturbed supersonic pulse. The perturbation grows relative to the unperturbed pulse as time advances; note the differing amplitude scales of the different panels. The pulse core is nearly restored without any phase shift at $t = 15$.
• Figure 2: Numerical evidence for the convective stability of kinks. Snapshots of perturbed kink ($c=0.9$) of (\ref{['eq: PDE in lab frame']})) in a reference frame of the same speed $c$. Perturbation at $t=0$ is concentrated to the right of its core. Red curves indicate the solution amplitude at different times. Blue dashed curves indicate the amplitude profile of the unperturbed kink. The perturbation departs from the kink core to its left. The kink core is nearly completely restored modulo a phase shift at $t = 60$; see Section \ref{['sec:nlstab-kink']}.
• Figure 3: Supersonic (dark and bright) pulses and their relations via discrete symmetries; Proposition \ref{['prop: discrete symmetries']}. Left panel: fixed $c > 1$. Right panel: fixed $c < -1$. Dark solid / dashed ellipses correspond to a particular $E$ such that $-1<E-c<1$. For example, referring to the left panel: suppose the solid blue curve is denoted $b_{c,E}$. Then, the dashed blue curve is $\mathcal{P} b_{c,E}=-b_{c,E}$. Further, the solid green curve is $\mathcal{T} b_{-c,-E}$, which travels with $c > 1$, and the dashed green curve is $\mathcal{PT} b_{-c,-E}$. The curves in the right panel arise by applying the transformation $\mathcal{T}$ to curves on the left panel, plotted with the same line colors and styles.
• Figure 4: Left panel: Orbit (solid curve), $b_{c,E}(x)=(u(x),v(x))$, in a phase portrait corresponding to a typical (dark) supersonic pulse; $c>1$ and $-1+c < E < 1+c$. Right panel: plot of components $u(x)$, $v(x)$ and its amplitude $r(x)=\sqrt{u^2(x)+v^2(x)}$.
• Figure 5: Kinks (solid), antikinks (dashed) and their relation through discrete symmetries. (a) $c \in [0,1)$ (b) $c \in (-1,0]$. In both of the plots, solid lines stand for kinks whose amplitudes $|b|$ increase in the same direction of their speed $c$; dashed lines are antikinks, whose amplitudes decrease against the direction of their speed $c$. Heteroclinic connections, which are interior to the unit circle and which connect points on the unit circle, which are not highlighted with solid curves, correspond to spatially bounded subsonic pulses. These are not stable. See section\ref{['sec: instability']}

Theorems & Definitions (49)

• Proposition 1.1: Discrete Symmetries
• Remark 1.2: Do solutions grow without bound in $L^\infty(\mathbb R)$?
• Proposition 2.1: Discrete symmetries of the family of traveling wave solutions
• Remark 2.2
• Proposition 3.1
• Theorem 3.2: Global well-posedness of the mild solution of (\ref{['eq: pert']})
• proof : Proof of Theorem \ref{['thm: global well-posedness perturbation H 1']}
• Proposition 3.3
• proof
• Theorem 4.1: Supersonic pulses are nonlinearly convectively stable