How close Are Integrable and Non-integrable Models: A Parametric Case Study Based on the Salerno Model

TL;DR

This study investigates how closely integrable and non-integrable models, specifically the Salerno model, resemble each other by examining the preservation of conserved quantities and the spectrum of Lyapunov exponents.

Contribution

It introduces the Salerno model as a parametric system to compare integrable and non-integrable dynamics and proposes the Lyapunov spectrum as a diagnostic tool for integrability breaking.

Findings

Conservation quantities are sensitive to deviations from integrability.

Lyapunov spectrum effectively indicates integrability breaking.

Even slight deviations significantly affect conserved quantities.

Abstract

In the present work we revisit the Salerno model as a prototypical system that interpolates between a well-known integrable system (the Ablowitz-Ladik lattice) and an experimentally tractable non-integrable one (the discrete nonlinear Schr\"odinger model). The question we ask is: for "generic" initial data, how close are the integrable to the non-integrable models? Our more precise formulation of this question is: how well is the constancy of formerly conserved quantities preserved in the non-integrable case? Upon examining this, we find that even slight deviations from integrability can be sensitively felt by measuring these formerly conserved quantities in the case of the Salerno model. However, given that the knowledge of these quantities requires a deep physical and mathematical analysis of the system, we seek a more "generic" diagnostic towards a manifestation of integrability breaking. We argue, based on our Salerno model computations, that the full spectrum of Lyapunov exponents could be a sensitive diagnostic to that effect.