String Equations of the q-KP Hierarchy

The q-deformed integrable system (also called q-analogue or q-deformation of classical integrable system) is defined by means of q-derivative ∂ q [1,2] instead of usual derivative ∂ with respect to x in a classical system. It reduces to a classical integrable system as q → 1. Recently, the q-deformed Kadomtsev-Petviashvili (q-KP) hierarchy is a subject of intensive study in the literature from [3] to [14]. Its infinite conservation laws, bi-Hamiltonian structure, τ function, additional symmetries and its constrained sub-hierarchy have already been reported in [4,5,11,12,14].

The additional symmetries, string equations and Virasoro constraints of the KP hierarchy are important as they are involved in the matrix models of the string theory [15]. For example, there are several new works [16][17][18][19][20] on this topic. The additional symmetries were discovered independently at least twice by Sato School [21] and Orlov-Shulman [22], in quite different environments and philosophy although they are equivalent essentially. It is well-known that L.A.Dickey [23] presented a very elegant and compact proof of Adler-Shiota-van Moerbeke (ASvM) formula [24,25] based on the Lax operator L and Orlov and Shulman’s M operator [22], and gave the string equation and the action of the additional symmetries on the τ function of the classical KP hierarchy. S.Panda and S.Roy gave the Virasoro and W -constraints on the τ function of the p-reduced KP hierarchy by expanding the additional symmetry operator in terms of the Lax operator [26,27]. It is quite interesting to study the analogous properties of q-deformed KP hierarchy by this expanding method . The main purpose of this article is to give the string equations of the q-KP hierarchy, and then study the negative Virasoro constraint generators {L -n , n ≥ 1} of 2-reduced q-KP hierarchy.

The organization of this paper is as follows. We recall some basic results and additional symmetries of q-KP hierarchy in Section 2. The string equations are given in Sections 3. The Virasoro constraints on the τ function of the 2-reduced (q-KdV) hierarchy are studied in Section 4. Section 5 is devoted to conclusions and discussions.

At the end of the this section, we shall collect some useful facts of q-calculus [2] to make this paper be self-contained. The q-derivative ∂ q is defined by

and the q-shift operator is

q denote the formal inverse of ∂ q . In general the following q-deformed Leibnitz rule holds

where the q-number and the q-binomial are defined by

For a q-pseudo-differential operator(q-PDO) of the form P = n i=-∞ p i ∂ i q , we separate P into the differential part P + = i≥0 p i ∂ i q and the integral part P -= i≤-1 p i ∂ i q . The conjugate operation " * " for P is defined by

q . The q-exponent e x q is defined as follows

Its equivalent expression is of the form

which is crucial to develop the τ function of the q-KP hierarchy [11].

2. q-KP hierarchy and its additional symmetries Similar to the general way of describing the classical KP hierarchy [21,28], we first give a brief introduction of q-KP hierarchy and its additional symmetries based on [11,12].

Let L be one q-PDO given by

which are called Lax operator of q-KP hierarchy. There exist infinite number of q-partial differential equations related to dynamical variables

and can be deduced from the generalized Lax equation,

which are called q-KP hierarchy. Here

in the following way

Dressing operator S satisfies Sato equation

The q-wave function w q (x, t; z) and the q-adjoint function w * q (x, t; z) are given by

which satisfies following linear q-differential equations

Furthermore, w q (x, t; z) and w * q (x, t; z) can be expressed by sole function τ q (x; t) [11] as

The following Lemma shows there exist an essential correspondence between q-KP hierarchy and KP hierarchy.

be a solution of the classical KP hierarchy and τ be its tau function. Then

is a tau function of the q-KP hierarchy associated with Lax operator L in eq. (2.1), where

Define Γ q and Orlov-Shulman’s M operator

(2.8) Eq. (2.2) together with eq. (2.8) implies that

(2.9)

Define the additional flows for each pair m, n as follows

or equivalently

(2.11)

(2.12)

The additional flows

commute with the hierarchy, i.e. [∂ * mn , ∂ k ] = 0 but do not commute with each other, so they are additional symmetries [12]. (M m L n ) -serves as the generator of the additional symmetries along the trajectory parametrized by t * m,n .

In this section we shall get string equations for the q-KP hierarchy from special additional symmetry flows. For this, we need a lemma. Proof. Direct calculations show that

= -1, where we have used [t i , ∂ q ] = 0 in the second step and

By virtue of Lemma 2, we have

The action of additional flows

which can be written as

3. The following theorem holds by virtue of eq.(3.3).

Theorem 1. If an operator L does not depend on the parameters t n and the additional variables t * 1,-n+1 , then L n is a purely differential oper

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