Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 110, 21 pages      arXiv:2408.09450      https://doi.org/10.3842/SIGMA.2024.110

The Modified Toda Hierarchy

Wenjuan Rui ab, Wenchuang Guan a, Yi Yang c and Jipeng Cheng ab
a) School of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu 221116, P.R. China
b) Jiangsu Center for Applied Mathematics (CUMT), Xuzhou, Jiangsu 221116, P.R. China
c) School of Mathematics, Sun Yat-sen University, Guangzhou, Guangdong 510000, P.R. China

Received August 20, 2024, in final form December 05, 2024; Published online December 11, 2024

Abstract
In this paper, modified Toda (mToda) equation is generalized to form an integrable hierarchy in the framework of Sato theory, which is therefore called mToda hierarchy. Inspired by the fact that Toda hierarchy is 2-component generalization of usual KP hierarchy, mToda hierarchy is constructed from bilinear equations of 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 Toda and mToda hierarchies, which means the definition of mToda hierarchy here is reasonable. Finally, Darboux transformations of the Toda and mToda hierarchies are also constructed by using the aforementioned Miura links.

Key words: modified Toda hierarchy; Toda hierarchy; Miura transformation; Darboux transformation; tau function.

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