Darboux transformations for two dimensional elliptic affine Toda equations

TL;DR

This paper develops Darboux transformations for all seven series of two-dimensional elliptic affine Toda equations, enabling the construction of exact solutions while preserving key symmetries.

Contribution

It introduces a uniform Darboux transformation framework for all seven series of elliptic affine Toda equations, maintaining their symmetries and providing explicit solutions.

Findings

Unified Darboux transformations for all series

Exact solutions constructed for each equation

Preservation of symmetry properties in solutions

Abstract

The Darboux transformations for the two dimensional elliptic affine Toda equations corresponding to all seven infinite series of affine Kac-Moody algebras, including $A_l^{(1)}$, $A_{2l}^{(2)}$, $A_{2l-1}^{(2)}$, $B_l^{(1)}$, $C_l^{(1)}$, $D_l^{(1)}$ and $D_{l+1}^{(2)}$, are presented. The Darboux transformation is constructed uniformly for the latter six series of equations with suitable choice of spectral parameters and the solutions of the Lax pairs so that all the reality symmetry, cyclic symmetry and complex orthogonal symmetry of the corresponding Lax pairs are kept invariant. The exact solutions of all these two dimensional elliptic affine Toda equations are obtained by using Darboux transformations.