Grupo de Física No Lineal, Departamento de Física Aplicada I, Universidad de Sevilla, Escuela Politécnica Superior, C/ Virgen de África, 7, 41011-Sevilla, Spain and Instituto de Matemáticas de la Universidad de Sevilla (IMUS), Edificio Celestino Mutis, Avda. Reina Mercedes s/n, 41012-Sevilla, Spain.
Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-9305, USA.
Phys Rev E. 2017 Sep;96(3-1):032214. doi: 10.1103/PhysRevE.96.032214. Epub 2017 Sep 15.
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 cotraveling 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 using 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 an 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.
在这项工作中，我们提供了两种互补的观点来研究哈密顿非线性动力晶格中孤波的（谱）稳定性，其中 Fermi-Pasta-Ulam 和 Toda 晶格是典型的例子。一种观点是将其视为在共行波坐标系下的定态解的特征值问题，另一种观点则是将其视为周期轨道的平移模。我们将前者的特征值与后者的 Floquet 乘子联系起来，并利用这个表述推导出一个基于能量的谱稳定性判据。它表明，当模型的能量（哈密顿量）H 对波速 c 的函数依赖性改变其单调性时，波稳定性的变化就会发生，这是一个充分（但不是必要）的条件。此外，在发生稳定性变化的临界速度附近，我们基于在临界速度 c_{0} 处评估的哈密顿量 H^{''}(c_{0})的二阶导数，提供了对不稳定特征值的一个显式的、一级近似的计算。我们用一系列可解析和数值处理的例子验证了这一结论，并讨论了它与最近基于能量的离散Breathers 稳定性判据的相似性。
