Sato theory on the q-Toda hierarchy and its extension

Abstract

In this paper, we construct the Sato theory including the Hirota bilinear equations and tau function of a new q-deformed Toda hierarchy(QTH). Meanwhile the Block type additional symmetry and bi-Hamiltonian structure of this hierarchy are given. From Hamiltonian tau symmetry, we give another definition of tau function of this hierarchy. Afterwards, we extend the q-Toda hierarchy to an extended q-Toda hierarchy(EQTH) which satisfy a generalized Hirota quadratic equation in terms of generalized vertex operators. The Hirota quadratic equation might have further application in Gromov-Witten theory. The corresponding Sato theory including multi-fold Darboux transformations of this extended hierarchy is also constructed. At last, we construct the multicomponent extension of the q-Toda hierarchy and show the integrability including its bi-Hamiltonian structure, tau symmetry and conserved densities.

Integrability on generalized q-Toda equation and hierarchy

“…Proof. The proof is similar as the proof in [2,8]. Here we will prove that the flows ∂ ∂t n are also Hamiltonian systems with respect to the first Poisson bracket.…”

“…The bi-Hamiltonian structure for the GQTH can be given by the following two compatible Poisson brackets similar as [2,8] {v…”

“…The Toda lattice equation is a completely integrable system which has many important applications in mathematics and physics including the theory of Lie algebra representation, orthogonal polynomials and random matrix model [3,19,20,23,24]. Toda system has many kinds of reduction or extension, for example extended Toda hierarchy (ETH) [2], bigraded Toda hierarchy (BTH) [1]- [8] and so on. These generalized Toda hierarchies have important application in Gromov-Witten theory on CP 1 and orbiford.…”

Bi-Hamiltonian structure of the extended noncommutative Toda hierarchy

“…Remark 4.3. The Lax equation and bi-hamiltonian structure of extended noncommutative Toda hierarchy will be reduced the ones of the extended Toda hierarchy in [11] when θ = Ω = 0, extended multicomponent Toda hierarchy in [24] when θ = Ω = 0 and g = gl N (C), extended Z N -Toda hierarchy in [25] when θ = Ω = 0 and the algebra takes values in a maximum commutative subalgebra of gl N (C), extended q-Toda hierarchy in [26] when θ = Ω = 0 and the algebra takes values in a q-shift algebra respectively.…”

Miura and auto-Backlund transformations for the q-deformed KP and q-deformed modified KP hierarchies

“…When q → 1, the q-deformed integrable systems will become into the classical systems. There are several kinds of q-deformed integrable systems, for example: the q-deformed Kadomtsev-Petviashvili (q-KP) hierarchy [1,3,[5][6][7][19][20][21][22], the q-deformed modified Kadomtsev-Petviashvili (q-mKP) hierarchy [4,13,18,23], the q-deformed AKNS-D hierarchy [24], the q-KP hierarchy and the q-mKP hierarchy with self-consistent sources [12,13], the q-Toda hierarchy [11] and so on. Among these q-deformed integrable systems, there is little literature on the q-mKP hierarchy.…”