Lucas polynomials and a standard Lax representation for the polytropic gas dynamics

TL;DR

This paper derives a standard Lax representation for polytropic gas dynamics using Lucas and Fibonacci polynomials, revealing conserved charges and extending to elastic medium equations, with discussions on dispersive extensions.

Contribution

It introduces a novel standard Lax representation for polytropic gas dynamics based on Lucas polynomials, connecting it with elastic medium equations and exploring dispersive extensions.

Findings

Derived a standard Lax representation for polytropic gas dynamics.

Identified conserved charges matching known non-standard representations.

Extended the framework to elastic medium equations and dispersive models.

Abstract

A standard Lax representation for the polytropic gas dynamics is derived by exploiting various properties of the Lucas and Fibonacci polynomials. The two infinite sets of conserved charges are derived from this representation and shown to coincide with the ones derived from the known non-standard representation. The same Lax function is shown to also give the standard Lax description for the elastic medium equations. In addition, some results on possible dispersive extensions of such models are presented.