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arXiv · 1701.04882

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

J. Cuevas-Maraver·P. G. Kevrekidis·A. Vainchtein·H. Xu
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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.

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J. Cuevas-Maraver, P. G. Kevrekidis, A. Vainchtein, H. Xu. 2017-08-22. A Unifying Perspective: Solitary Traveling Waves As Discrete Breathers And Energy Criteria For Their Stability. https://doi.org/10.1103/physreve.96.032214

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