Regular solution and lattice miura transformation of bigraded Toda Hierarchy

TL;DR

This paper develops finite-dimensional exponential solutions for the bigraded Toda Hierarchy, explores geometric structures, introduces a new Lax representation, and presents a lattice Miura transformation linking to equations like the Volterra lattice.

Contribution

It introduces a new Lax representation for the bigraded Toda Hierarchy and a lattice Miura transformation that simplifies the hierarchy to single-field equations.

Findings

Finite-dimensional exponential solutions for BTH.

A new Lax representation avoiding fractional operators.

Lattice Miura transformation connecting BTH to Volterra lattice.

Abstract

In this paper, we give finite dimensional exponential solutions of the bigraded Toda Hierarchy(BTH). As an specific example of exponential solutions of the BTH, we consider a regular solution for the $(1,2)$-BTH with $3\times 3$-sized Lax matrix, and discuss some geometric structure of the solution from which the difference between $(1,2)$-BTH and original Toda hierarchy is shown. After this, we construct another kind of Lax representation of $(N,1)$-bigraded Toda hierarchy($(N,1)$-BTH) which does not use the fractional operator of Lax operator. Then we introduce lattice Miura transformation of $(N,1)$-BTH which leads to equations depending on one field, meanwhile we give some specific examples which contains Volterra lattice equation(an useful ecological competition model).