The Fermi–Pasta–Ulam–Tsingou recurrence for discrete systems: Cascading mechanism and machine learning for the Ablowitz–Ladik equation

Abstract

The Fermi–Pasta–Ulam–Tsingou recurrence phenomenon for the Ablowitz–Ladik equation is studied analytically and computationally. Wave profiles periodic in the discrete coordinate may return to the initial states after complex stages of evolution. Theoretically this dynamics is interpreted through a cascading mechanism where higher order harmonics exponentially small initially grow at a faster rate than the fundamental mode. A breather is formed when all modes attain roughly the same magnitude. Numerically a fourth-order Runge–Kutta method is implemented to reproduce this recurring pattern. In another illuminating perspective, we employ data driven and machine learning techniques, e.g. back propagation, hidden physics and physics-informed neural networks. Using data from a fixed time as a learning basis, doubly periodic solutions in both the defocusing and focusing regimes are obtained. The predictions by neural networks are in excellent agreement with those from numerical simulations and analytical solutions.