https://iopscience.iop.org/article/10.3367/UFNe.2016.02.037729/pdf

Discrete breathers in crystals

We study the properties of spatially localized and time-periodic excitations—discrete breathers—in Fermi–Pasta–Ulam (FPU) chains. We provide a detailed analysis of their spatial profiles and stability properties. We especially demonstrate that the Page mode is linearly stable for symmetric FPU potentials. A resonant interaction between a localized and delocalized perturbations causes weak but finite strength instabilities for asymmetric FPU potentials. This interaction induces Fano resonances for plane waves scattered by the breather. Finally we analyze the interplay between energy thresholds for breathers in the presence of strongly asymmetric FPU potentials and the corresponding profiles of the low-frequency limit of breather families.

Discrete breathers in Fermi-Pasta-Ulam lattices

The Fermi-Pasta-Ulam (FPU) problem consists of the nonequipartition of energy among normal modes of a weakly anharmonic atomic chain model. In the harmonic limit each normal mode corresponds to a periodic orbit in phase space and is characterized by its wave number q. We continue normal modes from the harmonic limit into the FPU parameter regime and obtain persistence of these periodic orbits, termed here q-Breathers (QB). They are characterized by time periodicity, exponential localization in the q-space of normal modes and linear stability up to a size-dependent threshold amplitude. Trajectories computed in the original FPU setting are perturbations around these exact QB solutions. The QB concept is applicable to other nonlinear lattices as well.

/pdf/q-breathers-in-fermi-pasta-ulam-chains-existence-1ujbbiralp.pdf

q -breathers in Fermi-Pasta-Ulam chains: Existence, localization, and stability

Bushes of vibrational modes for Fermi–Pasta–Ulam chains

Tangent bifurcation of band edge plane waves, dynamical symmetry breaking and vibrational localization

A new type of finite-amplitude traveling or standing wave with an exact sinusoidal form and a short commensurate wavelength is predicted to exist in lattices with cubic and/or quartic anharmonic potential between any arbitrary number of nearest and non-nearest neighbors. Fast traveling nonlinear sinusoidal waves (NSW) can generate sinusoidal lattice solitons. Superposition of two NSW or sinusoidal solitons propagating in opposite directions can result in the formation of an extended or a localized standing-wave eigenmode. New exact solutions for localized standing-wave structures are found within a rigorous discrete-lattice approach.

/pdf/nonlinear-sinusoidal-waves-and-their-superposition-in-itwut0kf3v.pdf

Yuriy A. Kosevich

Papers

Nonlinear sinusoidal waves and their superposition in anharmonic lattices.