Modulational instability and resonant wave modes act on the metastability of oscillator chains

We describe the emergence and interactions of breather modes and resonant wave modes within a two-dimensional ringlike oscillator chain in a microcanonical situation. Our analytical results identify different dynamical regimes characterized by the potential dominance of either type of mode. The chain is initially placed in a metastable state, which it can leave by passing over the brim of the applied Mexican-hat-like potential. We elucidate the influence of the different wave modes on the mean-first passage time. A central finding is that also in this complex potential landscape a fast noise-free escape scenario solely relying on nonlinear cooperative effects is accomplishable even in a low-energy setting.

Figure 1

The left-hand side of the figure represents the profile of the Mexican-hat-like potential. The potential barrier vanishes for $\lambda >1$. The right-hand side of the figure shows the result of the stability analysis of the fixed point comprising the minimum-energy configuration and vanishing momenta for $N=100$ (see Appendix pp1 for details).

Figure 2

A typical initial preparation of the oscillator chain.

Figure 3

Simulation snapshots showing the growth of a radial breather array from an almost homogeneous initial state due to modulational instability. The ongoing amplification of this pattern eventually drives an individual oscillator over the potential barrier and thus triggers an escape of type I as described in Sec. 3. Parameters: $\kappa \Delta {\Theta }^2=0.79\times {10}^{-4}$, $\lambda =0.85$.

Figure 4

Simulation snapshots showing the emergence of a longitudinal wave pattern with wave number $m_{\phi }=5$. Arrows indicate the chain movement. Parameters: $\kappa \Delta {\Theta }^2=0.06$, $\lambda =0.4$.

Figure 5

Analytical results from the study of emerging wave modes. Symbols ($\blacksquare \hspace{0.17em}▴\hspace{0.17em}▾\hspace{0.17em}•$) indicate parameter choices according to Table 1. ${\epsilon }_{\mathrm{scale}}=4$. No wave modes are expected in black areas; gray parameter areas are either excluded (Fig. 1) or their activation energy could not be determined (Fig. 11).

Figure 6

The background plots depict how the total kinetic energy divides among the longitudinal (yellow) and radial (red) contribution, $E_{\mathrm{kin}}^r$ and $E_{\mathrm{kin}}^{\phi }$, as a function of time; see Eq. (13). The insets show the temporal evolution of absolute kinetic-energy values within the marked time frames. The three panels represent different parameter choices (indicated by the symbols  $\blacksquare \hspace{0.17em}▴\hspace{0.17em}•$ according to Table 1), which represent the three different dynamical regimes as described in the text. Time is measured in units of $k=0$ phonon periods $T_{k=0}=2\pi /{\omega }_0$. For all choices of parameters, ${\epsilon }_{\mathrm{scale}}=2$.

Figure 7

Relevant transition state types.

Figure 8

Snapshots of an escape process of type I. Parameters: $\lambda =0.8$, $\kappa \Delta {\Theta }^2=0.08$.

Figure 9

Snapshots of an escape process of type II. Arrows indicate the chain movement. Parameters: “$▴$” according to Table 1.

Figure 10

Comparison of the energies of different transition state types (white, no energy values determined; gray, excluded from parameter space).

Figure 11

Activation energy corresponds to ${\epsilon }_{\mathrm{scale}}=1$ (white, no energy values determined; gray, excluded from parameter space).

Figure 12

Cumulative escape time distributions. Parameters ($\blacksquare$: $m_r\approx 18$, $▴$: $m_{\phi }=2$, $▾$: $m_{\phi }=3$, $•$: disordered regime) according to Table 1.