Insensitive nonreciprocal edge breathers

TL;DR

The paper addresses how nonlinearity, nonreciprocity, and topology interact in a nonreciprocal Klein–Gordon chain to produce edge waves that remain robust against amplitude changes. Using multiple‑scale analysis and numerical continuation, the authors demonstrate continuous families of nonreciprocal edge breathers (NEBs) bifurcating from the linear edge mode, and identify a spectral insensitivity region (region II) where NEBs retain the linear edge‑mode frequency $oldsymbol{ extomega_0}$ despite nonlinearity. The insensitivity arises from a competition between edge‑mode nonorthogonality and nonlinear self‑interaction, yielding a nonlinear frequency shift that decays exponentially with lattice size $M$; this persists into the strongly nonlinear regime via avoided‑crossing bifurcations with skin‑mode families. The findings reveal a generic route to robust nonlinear topological waves in mechanical metamaterials without relying on symmetry‑protected nonlinearities, with potential applicability across mechanical, acoustic, and electronic platforms.

Abstract

We uncover subtle and previously unexplored phenomena arising from the interplay of nonlinearity and nonreciprocity in topological mechanical metamaterials. We study a nonreciprocal topological Klein-Gordon chain of asymmetrically coupled nonlinear oscillators, which serves as a minimal mass-spring model capturing the features of several active nonreciprocal metamaterials across mechanical, electronic, and acoustic platforms. We demonstrate that continuous families of nonreciprocal edge breathers (NEBs), namely boundary-localized, time-periodic waves, emerge from the linear edge mode as its amplitude increases. Remarkably, despite the absence of chiral or sublattice symmetries, we identify insensitive NEBs whose nonlinear frequency remains fixed to that of the linear edge mode with increasing nonlinearity. Our analysis reveals that the mechanism underlying this insensitivity stems from a competition between mode nonorthogonality and nonlinear interactions, yielding an exponential decay of the NEB nonlinear frequency shift with system size. Crucially, these insensitive NEBs also persist in the strongly nonlinear regime. Our work establishes a novel pathway toward realizing robust nonlinear topological waves in mechanical metamaterials without relying on symmetry-protected nonlinearities.

Figures (6)

• Figure 1: (a) Schematic of the unit cell of the nonreciprocal topological KG chain of nonlinear classical oscillators. The $s$ and $t\pm \gamma$ are the inter-and-intracell elastic stiffnesses, with $\gamma$ tuning the nonreciprocal strength. Here the elongation $\Delta y$ has the dimension of displacements. Further, the $\alpha$ and $g$ are the elastic and nonlinear stiffnessses of the onsite spring. (b) Dependence of the linear frequencies, $\omega_j$ against the nonreciprocal parameter, $\gamma$ numerically computed for a lattice of $N=33$ sites ($M=17$ cells) with $s=0.6$, see horizontal dashed line in (c). (c) Phase diagram of the nonreciprocal topological KG chain of classical oscillators. The phase diagram is obtained analyzing the localization and location of the right and left eigenvectors of the edge mode. The dashed line guides the eyes to $s=0.6$, with the star-, square- and dot-symbol indicating representative cases in regions (I), (II) and (III), with $\gamma=0.3$, $\gamma=0.6$ and $\gamma=0.825$ respectively, see text for details.
• Figure 2: Effects of softening nonlinearity ($g = -1$) on families of NEBs emerging from the edge mode with frequency $\omega_0 = 1.73$. In all cases, $s = 0.6$, $N = 33$, and $\alpha_0 = 1$. (a) Left panel: dependence of the nonlinear frequency $\Omega$ on the amplitude $\lVert \vec{y} \rVert^2$, for $\gamma = 0.3$ (dark blue) and $\gamma = 0.825$ (dark red), corresponding to regions (I) and (III), respectively. Right panel: dependence of the magnitude of the largest Floquet eigenvalue, $\max|\lambda|$, on $\lVert \vec{y} \rVert^2$ for the families in (a). (b) Profiles of the (left) displacement and (right) normal mode coordinates of representative NEBs with $\gamma = 0.3$ [dark-blue curve in (a)]. The bottom panels correspond to the NEB with $\lVert \vec{y} \rVert^2 = 0.076$ [green star in (a)], while the top panels correspond to the one with $\lVert \vec{y} \rVert^2 = 1$ [green dot in (a)]. (c) Same as (a), but for $\gamma = 0.6$ [dark blue, region (II)] and $\gamma = 0$ (dark red). (d) Same as (b), but for representative cases belonging to the family of NEBs with (II) $\gamma=0.6$ [cyan curve in (c)]. (e) Time evolution of representative NEBs with (bottom) $\lVert \vec{y} \rVert^2 = 0.076$ and (top) $\lVert \vec{y} \rVert^2 = 1$ for $\gamma = 0.3$ in region (I). The initial conditions for the numerical integration correspond to the profiles shown in (b), perturbed by a deviation vector $\delta y_n = 5 \times 10^{-3} y_n$. (f) Same as (e), but for the NEBs shown in (d). The arrows indicate dynamical instabilities.
• Figure 3: Effects of hardening nonlinearity ($g = 1$) on families of NEBs emerging from the edge mode with frequency $\omega_0 = 1.73$. In all cases, $s = 0.6$, $N = 33$, and $\alpha_0 = 1$. (a) Left panel: dependence of the nonlinear frequency $\Omega$ on the amplitude $\lVert \vec{y} \rVert^2$, for $\gamma = 0.3$ (dark blue) and $\gamma = 0.825$ (dark red), corresponding to regions (I) and (III), respectively. Right panel: dependence of the magnitude of the largest Floquet eigenvalue, $\max|\lambda|$, on $\lVert \vec{y} \rVert^2$ for the families in (a). (b) Profiles of the (left) displacement and (right) normal mode coordinates of representative NEBs for $\gamma = 0.3$ [dark-blue curve in (a)]. The bottom panels correspond to a representative NEB with $\lVert \vec{y} \rVert^2 = 0.039$ [green star in (a)], while the top panels correspond to the NEB with $\lVert \vec{y} \rVert^2 = 1$ [green dot in (a)]. (c) Same as (a), but for $\gamma = 0.6$ [dark blue, region (II)] and $\gamma = 0$ (dark purple). (d) Same as (b), but for representative cases belonging to the family of NEBs with (II) $\gamma=0.6$ [cyan curve in (c)]. (e) Time evolution of the representative NEBs with (bottom) $\lVert \vec{y} \rVert^2 = 0.039$ and (top) $\lVert \vec{y} \rVert^2 = 1$ for $\gamma = 0.3$ in region (I). The initial conditions for the numerical integration correspond to the profiles shown in (b), perturbed by a deviation vector $\delta y_n = 5 \times 10^{-3} y_n$. (f) Same as (e), but for the NEBs shown in (d). The arrows indicate dynamical instabilities.
• Figure 4: Numerically computed spectral sensitivity diagrams of the NEBs. (a) Dependence of the nonlinear frequency $\Omega$ of the NEBs on the nonreciprocity parameter $\gamma$. The calculations are performed at fixed amplitude $\lVert \vec{y} \rVert^2 = 0.1$ for a chain with $N = 33$, $\alpha_0 = 1$, and $s = 0.6$. Blue dot-connected symbols show the numerical results for hardening nonlinearity ($g = 1$), while red square-connected symbols correspond to softening nonlinearity ($g = -1$). The black curves represent the variation of the linear spectrum of the lattice, with the horizontal black line indicating the edge mode at frequency $\omega_0 = 1.73$ [see also Fig. \ref{['fig:chain_and_spectrum_lineear']}(b)]. (b) Dependence of the frequency sensitivity factor $\mathcal{S}_\Omega$ (see text for details) on the lattice size $M$ (total number of cells) for representative cases in regions (I) $\gamma = 0.3$, (II) $\gamma = 0.6$, and (III) $\gamma = 0.825$, displayed as blue triangle-, red square-, and green diamond-connected symbols, respectively. The straight lines guide the eye toward $\mathcal{S}_\Omega \sim e^{-\beta_\Omega M}$, with $\beta_\Omega = 0.7$ (dashed line) for (II) $\gamma = 0.6$ and $\beta_\Omega = 0$ (solid line) for (I) $\gamma = 0.3$ and (III) $\gamma = 0.825$, obtained from numerical fitting SMOOTH. (c) Dependence of the wave-function sensitivity factor $\mathcal{S}_y$ [Eq. \ref{['eq:wave_function_sensitivity_numerical']}] on the nonreciprocity parameter $\gamma$. The condition $\mathcal{S}_y = 0$ indicates that the NEB wave function coincides with a nonlinear edge mode. (d) Same as (b), but for $\mathcal{S}_y$. Numerical fitting yields $\mathcal{S}_y \sim e^{-\beta_y M}$ with $\beta_y = -0.3$ (solid line) for (I) $\gamma = 0.3$ and $\beta_y = -0.3$ (dashed line) for (II) $\gamma = 0.6$.
• Figure 5: Dependence of the modulus of the Floquet eigenvalues, $\lvert \lambda \rvert$, on the amplitude, $\lVert \vec{y}\rVert^2$, for (a) $N=19$ and (b) $N=41$, with $\alpha_0=1$, $g=-1$, $s=0.6$ and $\gamma=0.6$ (II). Representative Floquet spectra in the complex plane are shown for (c) $\lVert \vec{y}\rVert^2=0.49$ and (d) $\lVert \vec{y}\rVert^2=2$, in case of $N=19$. The red arrows guide the eye to instabilities.