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

## TL;DR

This paper offers a unified framework for analyzing the spectral stability of solitary traveling waves in Hamiltonian lattices, linking eigenvalue problems with energy criteria and providing explicit stability change conditions.

## Contribution

It introduces an energy-based spectral stability criterion for solitary waves in Hamiltonian lattices, connecting eigenvalues with energy dependence on wave velocity and providing explicit unstable eigenvalue calculations.

## Key findings

- Stability change linked to energy functional monotonicity.
- Explicit unstable eigenvalues near critical velocity.
- Validated criterion with analytical and numerical examples.

## 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 co-traveling 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 based on 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 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.