One reduction of the modified Toda hierarchy

2025/07/24 by Wang, Jinbiao, Guan, Wenchuang, Chen, Mengyao +1
#35Q51 #35Q53 #37K10 #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2507.18271

Abstract

The modified Toda (mToda) hierarchy is a two-component generalization of the 1-st modified KP (mKP) hierarchy, which connects the Toda hierarchy via Miura links and has two tau functions. Based on the fact that the mToda and 1-st mKP hierarchies share the same fermionic form, we firstly construct the reduction of the mToda hierarchy L₁(n)^M=L₂(n)^N+∑_l∈ℤ∑_i=1^mq_i,nΛ^lr_i,n+1Δ and (L₁(n)^M+L₂(n)^N)(1)=0, called the generalized bigraded modified Toda hierarchy, which can be viewed as a new two-component generalization of the constrained mKP hierarchy \mathfrakL^k=(\mathfrakL^k)_{≥ 1}+∑_i=1^m \mathfrakq_i∂^-1\mathfrakr_i∂. Next the relation with the Toda reduction L₁(n)^M=L₂(n)^N+∑_{l∈ ℤ}∑_i=1^mq_i,nΛ^lr_i,n is discussed. Finally we give equivalent formulations of the Toda and mToda reductions in terms of tau functions.

Citations

• Multi-component Toda lattice hierarchy
• Generalized Bigraded Toda Hierarchy
• Lax structure and tau function for large BKP hierarchy
• Multicomponent KP type hierarchies and their reductions, associated to conjugacy classes of Weyl groups of classical Lie algebras
• Equivalence of formulations of the MKP hierarchy and its polynomial\n tau-functions
• The geometry of spinors and the multicomponent BKP and DKP hierarchies

Related

• Generalized Bigraded Toda Hierarchy
• Lax structure and tau function for large BKP hierarchy
• The Extended Toda Hierarchy
• The Virasoro symmetries of the bigraded modified Toda hierarchy
• The constrained KP hierarchy and the bigraded Toda hierarchy of (M,1)-type