The Modified Toda Hierarchy - Rui, Wenjuan; Guan, Wenchuang; Yang, Yi; Cheng, Jipeng (Symmetry, Integrability and Geometry: Methods and Applications, 2024)
In this paper, the modified Toda (mToda) equation is generalized to form an integrable hierarchy in the framework of Sato theory, which is therefore called the mToda hierarchy. Inspired by the fact that the Toda hierarchy is a 2-component generalization of the usual KP hierarchy, the mToda hierarchy is constructed from bilinear equations of the 2-component first modified KP hierarchy, where we provide the corresponding equivalence with Lax formulations. Then it is demonstrated that there are Miura links between the Toda and mToda hierarchies, which means the definition of the mToda hierarchy here is reasonable. Finally, Darboux transformations of the Toda and mToda hierarchies are also constructed by using the aforementioned Miura links.