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In this work, we provide two complementary perspectives for the (spectral)stability of solitary traveling waves in Hamiltonian nonlinear dynamicallattices, of which the Fermi-Pasta-Ulam and the Toda lattice are prototypicalexamples. One is as an eigenvalue problem for a stationary solution in aco-traveling frame, while the other is as a periodic orbit modulo shifts. Weconnect the eigenvalues of the former with the Floquet multipliers of thelatter and based on this formulation derive an energy-based spectral stabilitycriterion. It states that a sufficient (but not necessary) condition for achange in the wave stability occurs when the functional dependence of theenergy (Hamiltonian) $H$ of the model on the wave velocity $c$ changes itsmonotonicity. Moreover, near the critical velocity where the change ofstability occurs, we provide explicit leading-order computation of the unstableeigenvalues, based on the second derivative of the Hamiltonian $H"(c_0)$evaluated at the critical velocity $c_0$. We corroborate this conclusion with aseries of analytically and numerically tractable examples and discuss itsparallels with a recent energy-based criterion for the stability of discretebreathers.

A Unifying Perspective: Solitary Traveling Waves As Discrete Breathers And Energy Criteria For Their Stability