Quasi-periodic breathers in Newton’s cradle

Abstract

We consider the parameterized Newton’s cradle lattice with Hertzian interactions in this paper. The positive parameters are { β n : | n| ≤ b} with a fixed integer b ≥ 0, and the Hertzian potential is [Formula: see text] for a fixed real number α > α* ≔ 12 b + 25. Corresponding to a large Lebesgue measure set of [Formula: see text], we show the existence of a family of small amplitude, linearly stable, quasi-periodic breathers for Newton’s cradle lattice, which are quasi-periodic in time with 2 b + 1 frequencies and localized in space with rate [Formula: see text] as | n| ≫ 1. To overcome obstacles in applying the Kolmogorov–Arnold–Moser (KAM) method due to the finite smoothness of V, especially when α is not an integer and to obtain a sharp estimate of the localization rate of the quasi-periodic breathers, the proof of our result uses the Jackson–Moser–Zehnder analytic approximation technique but with refined estimates on error bounds, depending on the smoothness and dimension, which provide crucial controls on the convergence of KAM iterations.

An infinite dimensional KAM theorem with normal degeneracy

“…Newton's cradle lattice is known as a simplified model for granular chains consisting of linear pendular and nonlinear interaction in form of Hertz's forces. The existence of (quasi-)periodic breathers and the corresponding stability were studied in [7,9,10]. The methods mentioned in these works are both numerical simulations and theoretical proof, such as KAM and Nash-Moser iterations.…”