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JOURNALS // Symmetry, Integrability and Geometry: Methods and Applications // Archive

SIGMA, 2024 Volume 20, 110, 21 pp. (Mi sigma2112)

This article is cited in 1 paper

The Modified Toda Hierarchy

Wenjuan Ruiab, Wenchuang Guanb, Yi Yangc, Jipeng Chengab

a Jiangsu Center for Applied Mathematics (CUMT), Xuzhou, Jiangsu 221116, P.R. China
b School of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu 221116, P.R. China
c School of Mathematics, Sun Yat-sen University, Guangzhou, Guangdong 510000, P.R. China

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.

Keywords: modified Toda hierarchy, Toda hierarchy, Miura transformation, Darboux transformation, tau function.

MSC: 35Q51, 37K10, 37K40

Received: August 20, 2024; in final form December 5, 2024; Published online December 11, 2024

Language: English

DOI: 10.3842/SIGMA.2024.110


ArXiv: 2408.09450


© Steklov Math. Inst. of RAS, 2025