Unifying perspective: Solitary traveling waves as discrete breathers in Hamiltonian lattices and energy criteria for their stability

In this work, we provide two complementary perspectives for the (spectral) stability of solitary traveling waves in Hamiltonian nonlinear dynamical lattices, of which the Fermi-Pasta-Ulam and the Toda lattice are prototypical examples. One is as an eigenvalue problem for a stationary solution in a cotraveling frame, while the other is as a periodic orbit modulo shifts. We connect the eigenvalues of the former with the Floquet multipliers of the latter and using this formulation derive an energy-based spectral stability criterion. It states that a sufficient (but not necessary) condition for a change in the wave stability occurs when the functional dependence of the energy (Hamiltonian) $H$ of the model on the wave velocity $c$ changes its monotonicity. Moreover, near the critical velocity where the change of stability occurs, we provide an explicit leading-order computation of the unstable eigenvalues, based on the second derivative of the Hamiltonian $H^{\prime \prime }\left(c_0\right)$ evaluated at the critical velocity $c_0$. We corroborate this conclusion with a series of analytically and numerically tractable examples and discuss its parallels with a recent energy-based criterion for the stability of discrete breathers.

Figure 1

Top panel: Dependence of the energy $H$ on the wave velocity $c$ in the $\alpha$-FPU model in Eq. (9) with $\mathrm{\Lambda }\left(m\right)=0$. Bottom panels: Typical profile of the traveling wave with $c=1.5$ in the displacement ($u_n$) and strain ($y_n=u_{n+1}-u_n$) variables.

Figure 2

Stability and instability of the lattice traveling waves in the $\alpha$-FPU lattice with nearest-neighbor interactions governed by the potential in Eq. (9) and Kac-Baker long-range interactions with the kernel in Eq. (8). Here $\gamma =0.17$ and $\rho =0.0172$. The top panel shows the energy dependence on the speed, with $H^{\prime }\left(c\right)>0$ implying (spectral) stability, and $H^{\prime }\left(c\right)<0$ implying instability. The bottom panel confirms this by showing the maximum real eigenvalue obtained by diagonalizing the linearization operator $\mathcal{L}$ (dots) and transforming the relevant Floquet multiplier $\mu$ into a corresponding eigenvalue (for comparison) via the relation $\lambda =clog\left(\mu \right)$ (solid curve). The inset of the bottom panel shows the dependence of ${\lambda }^2$ on $c-c_0$ for $c$ near $c_0=1.6937$, the location of the second bifurcation; it fits a straight line ${\lambda }^2=\beta (c-c_0)$, with $\beta =-3.0383\times {10}^{-4}$.

Figure 3

Stability and instability of the lattice traveling waves of the model of Ref. [14] with the potential in Eq. (10). Here $\chi =4$ and $u_c=1$. The top panel displays the $H\left(c\right)$ dependence, which possesses a minimum at $c=c_0=1.0493$. The bottom panels show the dependence of the energy dispersion ${\varepsilon }_{\infty }$ and relative velocity change ${\eta }_{\infty }$ (see the text) with respect to the velocity $c$, which manifest the instability of solitary waves with $c<c_0$, where $c_0$ is such that $H^{\prime }\left(c_0\right)=0$.

Figure 4

Evolution of unstable traveling waves in the model of Ref. [14] with the potential in Eq. (10). Here $\chi =4$ and $u_c=1$. The panels show the profile of the strains $y_n\left(t\right)=u_{n+1}\left(t\right)-u_n\left(t\right)$ at $t=1500$ and zooms in the space-time evolution dynamics of the strains are represented in the insets. Top and bottom panels correspond to $c=1.025$ and $c=1.034$, respectively. In the the example shown in the bottom panel, the velocity eventually oscillates in time around an average value of 1.0626.