Infinite-dimensional 3-algebra and integrable system

TL;DR

This paper explores the connection between infinite-dimensional 3-algebras and the dispersionless KdV hierarchy, revealing bi-Hamiltonian and bi-Nambu-Hamiltonian structures, and applying these to integrable gas dynamics systems.

Contribution

It introduces a novel framework linking infinite-dimensional 3-algebras with integrable hierarchies via Nambu mechanics, expanding understanding of Hamiltonian structures in such systems.

Findings

Dispersionless KdV hierarchy derived from Nambu-Poisson structures.

Dispersionless KdV system is both bi-Hamiltonian and bi-Nambu-Hamiltonian.

Application to integrable gas dynamics systems.

Abstract

The relation between the infinite-dimensional 3-algebras and the dispersionless KdV hierarchy is investigated. Based on the infinite-dimensional 3-algebras, we derive two compatible Nambu Hamiltonian structures. Then the dispersionless KdV hierarchy follows from the Nambu-Poisson evolution equation given the suitable Hamiltonians. We find that the dispersionless KdV system is not only a bi-Hamiltonian system, but also a bi-Nambu-Hamiltonian system. Due to the Nambu-Poisson evolution equation involving two Hamiltonians, more intriguing relationships between these Hamiltonians are revealed. As an application, we investigate the system of polytropic gas equations and derive an integrable gas dynamics system in the framework of Nambu mechanics.