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 {βn : |n| ≤ b} with a fixed integer b ≥ 0, and the Hertzian potential is V(x)=11+α|x|1+α for a fixed real number α > α* ≔ 12b + 25. Corresponding to a large Lebesgue measure set of (βj)|j|≤b∈R+2b+1, 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 2b + 1 frequencies and localized in space with rate 1|n|1+α 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.