Stability of traveling waves in a driven Frenkel–Kontorova model

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 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.

Breathers in lattices with alternating strain-hardening and strain-softening interactions

“…9, the eigenvalues at the first turning point (i.e., a d ≈ 0.1242) are λ 1 ≈ 0.9992 and λ 2 ≈ 0.9976. Interestingly and in line with similar earlier observations in [57], they satisfy the relation λ 1 λ 2 = e −γT d which for the Hamiltonian case, i.e., γ = 0, it gives λ 1 λ 2 = 1. Indeed, we have λ 1 λ 2 ≈ 0.9968 and e −γT d ≈ 0.9968.…”

Simulation of Epitaxial Film–Substrate Interaction Potential

“…The difference between this potential and the exact value (7) does not exceed 1.5%. Potential (8) coincides with Frenkel-Kontorova potential up to the constant and linear transformation of coordinates x = x + 1/2.…”

“…This range is 0.8 ≤ σ ≤ 1.2 for metals. The first two terms dominate the rest in the Fourier series for potential (7) at σ > 0.9. The potential is approximated by a simple formula in this case.…”

“…The cosine function specifies the periodic substrate potential. This simple approximation reveals a wide variety of ground states for the system [4][5][6] and different waves in the atoms chain [7][8][9]. Solitons are of the greatest interest.…”