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.

在这项工作中，我们将重新讨论由恒定外力驱动的阻尼Frenkel-Kontorova晶格中行波的经典问题。我们将这些解计算为非线性映射的固定点，并针对不同的阻尼系数值获得驱动力和波速之间的对应动力学关系。我们表明，由于线性模式的共振，动力学曲线在小速度下会变为非单调，在大速度下动力学关系变为多值。探索获得的波形的频谱稳定性，我们在计算的数值精度水平上，确定行波解的不稳定性的精确标准：动力学曲线的单调递减部分始终具有不稳定的本征方向。我们讨论了为什么该准则在耗散环境中的有效性是一个相当显着的特征，从而提供了与模型的汉密尔顿变体以及更普遍的晶格行波的联系。我们的稳定性结果通过直接数值模拟得到了证实，这些数值模拟还揭示了动态不稳定性的可能结果。