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

Abstract

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 (c 0 ) 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.

Stability of traveling waves in a driven Frenkel–Kontorova model

“…Thirdly, this criterion is similar in spirit with the criteria that we have established in our recent works in [21,22] (and with the classic work of [34]). We have, in fact, attempted to extract such a "theorem" following the pattern of the corresponding proofs in [34,21,22]. However, this attempt encounters two significant stumbling blocks.…”

“…This, in turn, allowed a definitive computation of the Floquet multipliers associated with the solution considered as a periodic orbit (modulo shifts). We have argued both here and in our earlier works [21,22] that this is a beneficial way of examining traveling wave solutions on a lattice as it helps to understand their stability and hence provides an informed view on their dynamics.…”

“…Hence, the very core of the calculation (see, for example, the line between Eqs. (5) and (6) in [21]) is no longer immediately valid.…”

An energy-based stability criterion for solitary travelling waves in Hamiltonian lattices

“…More specifically, we showed that when H ′ (c 0 ) = 0 for some critical velocity c 0 , a pair of eigenvalues associated with the traveling wave (in the co-traveling frame of reference in which the wave is steady) cross zero and emerge on the real axis when c is either above or below c 0 , thus creating instability. While this energy-based criterion first appeared in [16], where it was motivated by the study of the FPU problem in the near-sonic limit, in [31] we provided a concise proof as well as an explicit leading-order calculation for these two near-zero eigenvalues. In addition, we tested the criterion numerically for a range of lattice problems.…”

“…In a recent paper [31], inspired by and extending the fundamental work of [16], we examined the sufficient (but not necessary) condition for a change in the wave's spectral stability occurring when the function H(c) changes its monotonicity. More specifically, we showed that when H ′ (c 0 ) = 0 for some critical velocity c 0 , a pair of eigenvalues associated with the traveling wave (in the co-traveling frame of reference in which the wave is steady) cross zero and emerge on the real axis when c is either above or below c 0 , thus creating instability.…”