The Lax operator fixed under the additional symmetries of the extended Toda hierarchy

Additional symmetries of the extended bigraded Toda hierarchy were introduced by Bakalov and Wheeless. We describe properties of a Lax operator fixed under these additional symmetries and determine the unique such Lax operator in the special case of the extended Toda hierarchy (ETH). We further present differential equations for the wave functions and their general solutions that hint at a possible connection to the bispectral problem. Based on these solutions, we highlight a correspondence between the wave functions. Finally, we find the form of a tau function for our Lax operator and compute a second solution to the ETH by applying a Darboux transformation presented by Li and Song.

Automated summary

This article introduces the additional symmetries of the extended bigraded Toda hierarchy and the properties of the Lax operator fixed under these symmetries. It determines the unique Lax operator in the special case of the extended Toda hierarchy and presents the differential equations and general solutions for the wave functions, suggesting a possible connection to the bispectral problem. Moreover, the article finds the form of the tau function for the Lax operator and computes a second solution to the extended Toda hierarchy using a Darboux transformation.