Differential equations for the closed geometric crystal chains

TL;DR

This paper introduces two differential equation systems derived from discrete integrable systems called closed geometric crystal chains, including an extended Lotka-Volterra system and a new system with Lax representations linked to loop elementary symmetric functions.

Contribution

It presents a novel method for deriving continuous Lax equations from discrete systems using Puiseux series expansions of eigenvalues.

Findings

Derived differential equations from discrete integrable systems.

Connected Lax representations to loop elementary symmetric functions.

Introduced a new differential system reducing to known models in special cases.

Abstract

We present two types of systems of differential equations that can be derived from a set of discrete integrable systems which we call the closed geometric crystal chains. One is a kind of extended Lotka-Volterra systems, and the other seems to be generally new but reduces to a previously known system in a special case. Both equations have Lax representations associated with what are known as the loop elementary symmetric functions, which were originally introduced to describe products of affine type A geometric crystals for symmetric tensor representations. Examples of the derivations of the continuous time Lax equations from a discrete time one are described in detail, where a novel method of taking a continuum limit by assuming asymptotic behaviors of the eigenvalues of the Lax matrix in Puiseux series expansions is used.