Linearization, separability and Lax pairs representation of $a_4^{(2)}$ Toda lattice
Bruce Lionnel Lietap Ndi, Djagwa Dehainsala, Joseph Dongho
TL;DR
This work addresses the linearization and Lax-pair representation of the two-dimensional Toda lattice associated with the twisted affine Lie algebra $a_4^{(2)}$ within the algebraic complete integrability framework. It demonstrates that generic invariant fibers complete to an abelian surface isomorphic to the Jacobian of a genus-2 hyperelliptic curve, enabling linearization on the Jacobian via spectral data. By constructing an explicit morphism to the Mumford system, the authors derive a new Poisson structure and a Lax equation with a spectral parameter for the Toda lattice, connecting the Kummer and Jacobian geometries to theta-function linearization. Overall, the paper provides a concrete, computable route from the Toda dynamics to Abelian-variety linearization and a corresponding Lax formulation, enhancing understanding of $a_4^{(2)}$ integrable structures and their geometric underpinnings.
Abstract
The aim of this work is focused on linearizing and found the Lax Pairs of the algebraic complete integrability (a.c.i) Toda lattice associated with the twisted affine Lie algebra \(a_4^{\left(2\right)}\). Firstly, we recall that our case of a.c.i is a two-dimensional algebraic completely integrable systems for which the invariant (real) tori can be extended to complex algebraic tori (abelian surfaces). This implies that the geometry can be used to study this system. Secondly, we show that the lattice is related to the Mumford system and we construct an explicit morphism between these systems, leading to a new Poisson structure for the Mumford system. Finally, we give a new Lax equation for this Toda lattice and we construct an explicit linearization of the system.