Novikov algebras and a classification of multicomponent Camassa-Holm equations

TL;DR

This paper classifies low-dimensional Novikov algebras and constructs multi-component integrable systems, including known equations, that interpolate between KdV and Camassa-Holm types, revealing new bi-Hamiltonian structures.

Contribution

It introduces a classification of Novikov algebras and constructs new multi-component integrable systems based on this classification.

Findings

Classified low-dimensional Novikov algebras.

Constructed multi-component bi-Hamiltonian systems.

Presented examples including known integrable equations.

Abstract

A class of multi-component integrable systems associated to Novikov algebras, which interpolate between KdV and Camassa-Holm type equations, is obtained. The construction is based on the classification of low-dimensional Novikov algebras by Bai and Meng. These multi-component bi-Hamiltonian systems obtained by this construction may be interpreted as Euler equations on the centrally extended Lie algebras associated to the Novikov algebras. The related bilinear forms generating cocycles of first, second and third order are classified. Several examples, including known integrable equations, are presented.