Lotka--Volterra Type Equations and their Explicit Integration

TL;DR

This paper provides explicit solutions for certain two-dimensional Lotka--Volterra equations linked to simple Lie algebras, extending methods beyond the well-known $A_n$ case to other types like $B_r$, $C_r$, and $G_2$, unifying their treatment.

Contribution

It introduces explicit integration techniques for Lotka--Volterra equations associated with simple Lie algebras beyond $A_n$, enabling a unified approach for multiple algebra types.

Findings

Explicit solutions for non-$A_n$ Lie algebra cases

Unified treatment of first fundamental representations

Extension of integrability methods to broader algebra classes

Abstract

In the present note we give an explicit integration of some two--dimensionalised Lotka--Volterra type equations associated with simple Lie algebras, other than the familiar $A_n$ case, possessing a representation without branching. This allows us, in particular, to treat the first fundamental representations of $A_r$, $B_r$, $C_r$, and $G_2$ on the same footing.