Stability of traveling waves in a driven Frenkel-Kontorova model

TL;DR

This paper investigates the stability and bifurcation properties of traveling waves in a damped driven Frenkel-Kontorova lattice, revealing conditions for stability and instability linked to the shape of the kinetic relation.

Contribution

It introduces a numerical method to compute traveling wave solutions as fixed points and establishes a stability criterion based on the kinetic curve's monotonicity.

Findings

Kinetic relation can become non-monotone due to resonances.

Monotonically decreasing kinetic curve segments are unstable.

Numerical simulations confirm the stability criterion and dynamical outcomes.

Abstract

In this work we revisit a classical problem of traveling waves in a damped Frenkel-Kontorova lattice driven by a constant external force. We compute these solutions as fixed points of a nonlinear map and obtain the corresponding kinetic relation between the driving force and the velocity of the wave for different values of the damping coefficient. We show that the kinetic curve can become non-monotone at small velocities, due to resonances with linear modes, and also at large velocities where the kinetic relation becomes multivalued. Exploring the spectral stability of the obtained waveforms, we identify, at the level of numerical accuracy of our computations, a precise criterion for instability of the traveling wave solutions: monotonically decreasing portions of the kinetic curve always bear an unstable eigendirection. We discuss why the validity of this criterion in the {\it dissipative} setting is a rather remarkable feature offering connections to the Hamiltonian variant of the model and of lattice traveling waves more generally. Our stability results are corroborated by direct numerical simulations which also reveal the possible outcomes of dynamical instabilities.