Integrable Hierarchies and Information Measures

TL;DR

This paper explores the connection between integrable models and information theory, deriving new measures and equations such as a semi-relativistic NLS, highlighting the interplay between physics and information measures.

Contribution

It introduces higher order information measures within the integrable hierarchy of the NLS model, linking them to known measures and deriving a semi-relativistic NLS equation.

Findings

Higher order information measures include Fisher and Kullback-Leibler measures.

A Wootters type measure is derived from the NLS hierarchy.

An integrable semi-relativistic NLS equation is constructed.

Abstract

In this paper we investigate integrable models from the perspective of information theory, exhibiting various connections. We begin by showing that compressible hydrodynamics for a one-dimesional isentropic fluid, with an appropriately motivated information theoretic extension, is described by a general nonlinear Schrodinger (NLS) equation. Depending on the choice of the enthalpy function, one obtains the cubic NLS or other modified NLS equations that have applications in various fields. Next, by considering the integrable hierarchy associated with the NLS model, we propose higher order information measures which include the Fisher measure as their first member. The lowest members of the hiearchy are shown to be included in the expansion of a regularized Kullback-Leibler measure while, on the other hand, a suitable combination of the NLS hierarchy leads to a Wootters type measure related to a NLS equation with a relativistic dispersion relation. Finally, through our approach, we are led to construct an integrable semi-relativistic NLS equation.