Particle-like, dyx-coaxial and trix-coaxial Lie algebra structures for a multi-dimensional continuous Toda type system

TL;DR

This paper explores algebraic structures called Lie algebras associated with a 2+1-dimensional Toda system, revealing new particle-like and coaxial algebraic configurations with applications to conservation laws and spectral problems.

Contribution

It introduces novel Lie algebra structures (dyx-coaxial and trix-coaxial) linked to the Toda system, expanding the algebraic understanding of such integrable models.

Findings

Identification of particle-like Lie algebra structures for the Toda system

Construction of trix-coaxial and dyx-coaxial Lie algebra structures

Applications to conservation laws and inverse spectral problems

Abstract

We prove that with a $(2+1)$-dimensional Toda type system are associated algebraic skeletons which are (compatible assemblings) of particle-like Lie algebras of dyons and triadons type. We obtain trix-coaxial and dyx-coaxial Lie algebra structures for the system from algebraic skeletons of some particular choice for compatible associated absolute parallelisms. In particular, by a first choice of the absolute parallelism, we associate with the $(2+1)$-dimensional Toda type system a trix-coaxial Lie algebra structure made of two (compatible) base triadons constituting a $2$-catena. Furthermore, by a second choice of the absolute parallelism, we associate a dyx-coaxial Lie algebra structure made of two (compatible) base dyons, as well as particle-like Lie algebra structures made of single $3$-dyons. Some explicit examples of applications such as conservation laws related to special solutions, and an inverse spectral problem are worked out.