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

TL;DR

This paper revisits an energy-based criterion for the stability of solitary traveling waves in Hamiltonian lattices, analyzing its implications through spectral and Floquet theory, and corroborates predictions with numerical results in specific models.

Contribution

It provides a detailed analysis of the stability criterion using spectral and Floquet perspectives, establishing their correspondence and explicit velocity dependence near critical points.

Findings

The stability criterion's eigenvalue and Floquet multiplier are explicitly related.

Numerical results confirm the criterion's predictions in two models.

The stability may change twice as the wave velocity varies.

Abstract

In this work, we revisit a criterion, originally proposed in [Nonlinearity {\bf 17}, 207 (2004)], for the stability of solitary traveling waves in Hamiltonian, infinite-dimensional lattice dynamical systems. We discuss the implications of this criterion from the point of view of stability theory, both at the level of the spectral analysis of the advance-delay differential equations in the co-traveling frame, as well as at that of the Floquet problem arising when considering the traveling wave as a periodic orbit modulo a shift. We establish the correspondence of these perspectives for the pertinent eigenvalue and Floquet multiplier and provide explicit expressions for their dependence on the velocity of the traveling wave in the vicinity of the critical point. Numerical results are used to corroborate the relevant predictions in two different models, where the stability may change twice. Some extensions, generalizations and future directions of this investigation are also discussed.