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

Abstract

<p>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 <a href="https://www.sciencedirect.com/topics/engineering/machine-learning-technique" title="Learn more about machine learning techniques from ScienceDirect's AI-generated Topic Pages">machine learning techniques</a>, e.g. <a href="https://www.sciencedirect.com/topics/engineering/backpropagation" title="Learn more about back propagation from ScienceDirect's AI-generated Topic Pages">back propagation</a>, hidden <a href="https://www.sciencedirect.com/topics/physics-and-astronomy/physics" title="Learn more about physics from ScienceDirect's AI-generated Topic Pages">physics</a> and physics-informed <a href="https://www.sciencedirect.com/topics/mathematics/neural-network" title="Learn more about neural networks from ScienceDirect's AI-generated Topic Pages">neural networks</a>. 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 <a href="https://www.sciencedirect.com/topics/mathematics/neural-network" title="Learn more about neural networks from ScienceDirect's AI-generated Topic Pages">neural networks</a> are in excellent agreement with those from numerical simulations and analytical solutions.<br></p>