Solutions of a discretized Toda field equation for $D_{r}$ from Analytic Bethe Ansatz

TL;DR

This paper derives explicit solutions for a discretized Toda field equation associated with the $D_r$ algebra, using analytic Bethe ansatz to express solutions via determinants and Pfaffians, advancing integrable systems theory.

Contribution

It introduces a novel determinant and Pfaffian-based solution for the $D_r$ Toda field equation derived from the analytic Bethe ansatz, specific to the $D_r$ case.

Findings

Explicit determinant and Pfaffian solutions for the $D_r$ Toda equation.

Connection between transfer matrix relations and Toda field equations.

Advancement in solving discrete integrable systems using Bethe ansatz.

Abstract

Commuting transfer matrices of $U_{q}(X_{r}^{(1)})$ vertex models obey the functional relations which can be viewed as an $X_{r}$ type Toda field equation on discrete space time. Based on analytic Bethe ansatz we present, for $X_{r}=D_{r}$, a new expression of its solution in terms of determinants and Pfaffians.