The Successive Applications of Two Types of Gauge Transformations for the q-Deformed Modified Kadomtsev-Petviashvili Hierarchy

Na Li; Jipeng Cheng

doi:10.1515/zna-2017-0433

Article Publicly Available

The Successive Applications of Two Types of Gauge Transformations for the q-Deformed Modified Kadomtsev-Petviashvili Hierarchy

Na Li and Jipeng Cheng

Published/Copyright: February 15, 2018

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From the journal Zeitschrift für Naturforschung A Volume 73 Issue 4

Abstract

In this paper, the successive applications of two types of gauge transformation T_D and TI for the q-deformed modified Kadomtsev-Petviashvili hierarchy are discussed. It is found that T_D and T_I can commute with each other. We mainly studied products of n terms of T_D and k terms of T_I in three cases for different values of n and k. Finally, some applications of these results are also given.

Keywords: Gauge Transformation; q-mKP Hierarchy; Successive Applications

PACS: 02.30.Ik

MSC 2010: 35Q53; 37K10; 37K40

1 Introduction

Recently, much attention is paid to the q-deformed integrable system [1], [2], [3], which is one of the most significant research objects in mathematical physics. It is defined by using the q-derivative ∂_q [1], [2], [3] to replace common derivative ∂_x in the classical system. Many kinds of q-deformed integrable systems were studied, such as the q-deformed Kadomtsev-Petviashvili (q-KP) hierarchy [1], [3], [4], [5], [6], [7], the q-deformed modified Kadomtsev-Petviashvili (q-mKP) hierarchy [8], [9], [10], [11], [12], the q-KP hierarchy and the q-mKP hierarchy with self-consistent sources [11], [13], q-Toda hierarchy [14], q-AKNS-D hierarchy [15] and so on. In this paper, we will mainly investigate the q-mKP hierarchy.

The gauge transformation [16], [17] is also an important object as it offers a convenient method to solve integrable hierarchies. Until now, the gauge transformations of many integrable hierarchies studied were, for example, the KP and mKP hierarchies [16], [17], [18], [19], [20], [21], the discrete KP hierarchy [22], the q-KP and q-mKP hierarchies [3], [8] and so on. Here, we will focus on the gauge transformation of the q-mKP hierarchy.

Even though the gauge transformation of the q-mKP hierarchy was partially discussed in [8], the investigation was not complete. There is a lack of studies on the gauge transformations generated by the adjoint eigenfunctions in [8]. Hence, we will continue to study the gauge transformation of the q-mKP hierarchy in this article. It is found that there are three kinds of elementary gauge transformation operators for the q-mKP hierarchy, i.e. T_i with i=1, 2, 3 (see Sec. 3). But T_i with i=1, 2, 3 cannot commute with each other, and thus they are not convenient in the application. Therefore two types of gauge transformation operators T_D and T_I [9] constructed from T_i (see Sec. 4) were introduced and it is proved that they can commute with each other, so they are more applicable in the q-mKP hierarchy than T_i with i=1, 2, 3. Then, for the successive applications of n terms of T_D and k terms of T_I, we study three cases of n>k, n=k and n<k in the q-mKP hierarchy. Finally, the application of T^(n,k) on the eigenfunction and adjoint eigenfunction of the q-mKP hierarchy is studied.

This paper is organised in the following way. In Section 2, some fundamental facts on the q-mKP hierarchy are reviewed. Next, we discuss the successive applications and the commutativity of the elementary gauge transformation operators T_i with i=1, 2, 3 in Section 3. Then the successive applications of the gauge transformation operators T_D(Φ) and T_I(Ψ) are studied in three cases for different n and k values in Section 4. In Section 5, some applications of the corresponding results are summarised. Finally, we give some conclusions and discussions in Section 6.

2 The q-mKP Hierarchy

The q-derivative ∂_q and the q-shift operator are defined by their actions on a function f(x) [1], [2], [3]

(1)∂q(f(x))=f(qx)−f(x)x(q−1), θ(f(x))=f(qx).

Let ∂q−1 denote the formal inverse of ∂_q, we note that θ does not commute with ∂_q,

(2)∂q(θk(f))=qkθk(∂q(f)), k∈Z.

And the algebraic multiplication of ∂qi with the multiplication operator f is given by the q-deformed Leibnitz rule below [1], [2], [3]

(3)∂qif=∑j≥0(ij)qθi−j(∂qj(f))∂qi−j, i∈ℤ,

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

(4)(nk)q=(n)q(n−1)q⋯(n−k+1)q(1)q(1)q⋯(k)q, (n)q=qn−1q−1, (n0)q=1.

For A=Σiai∂qi, we denote A≥k=Σi≥kai∂qi and A<k=Σi<kai∂qi. What’s more, A(f) indicates the action of A on f, while Af or A·f denotes the multiplication of A and f, and ∗ stands for the conjugate operation: (AB)^*=B^*A^*, ∂q∗=−∂qθ−1=−1q∂1q,(∂q−1)∗=(∂q∗)−1=−θ∂q−1,f^*=f. Define the q-mKP hierarchy [8], [9], [11] as the following Lax equation

(5)Ltn=[(Ln)≥1, L], n=1, 2, 3, ⋯

with the Lax operator L given by the q-pseudo-differential operator below

(6)L=v1∂q+v0+v−1∂q−1+v−2∂q−2+v−3∂q−3+⋯

Here v_i=v_i(x, t)=v_i(x, t₁, t₂, L).

The first nontrivial equation of the q-mKP hierarchy is the q-mKP equation [9]

(7)∂v∂t2=∂2v∂t12+∂q(v)∂v∂t1+(2v∂∂t1−vθ(v)∂q−v2∂q+2∂v∂t1)((θ−1)−11v∂v∂t1).

where (θ−1)⁻¹ means −Σi≥0θi.

The Lax operator L for the q-mKP hierarchy can be expressed in terms of the dressing operator Z [9],

(8)L=Z∂qZ−1,

where Z is given by

(9)Z=z0+z1∂q−1+z2∂q−2+⋯ (z0−1 exists).

Then the Lax equation (5) is equivalent to

(10)∂Z∂tn=−(Ln)≤0Z=−(Z∂qnZ−1)≤0Z.

The eigenfunction Φ and the adjoint eigenfunction Ψ of the q-mKP hierarchy are defined by the following equations

(11)∂Φ∂tn=(Ln)≥1(Φ), ∂Ψ∂tn=−(Ln)≥1∗(Ψ).

In paticular, 1 is the eigenfunction.

3 Elementary Gauge Transformation

For the q-mKP hierarchy (5), suppose T is a q-pseudo-differential operator, and

(12)L(1)=TLT−1,

such that

(13)Ltn(1)=[(L(1))≥1n, L(1)]

still holds for the transformed Lax operator L⁽¹⁾, then we call T a gauge transformation operator of the q-mKP hierarchy. According to (13), the following lemma can be easily obtained.

Lemma 1.If the q-pseudo-differential operator T satisfies[8]

(14)(TLnT−1)≥1=T(Ln)≥1T−1+TtnT−1,

then T is a gauge transformation operator of the q-mKP hierarchy.

Before the discussion of the gauge transformation, the following basic identities on the q-pseudo-differential operator are needed.

Lemma 2.For any q-pseudo-differential operator A and arbitrary functions f, f₁, f₂, g₁and g₂, the following operator identities hold:

(15)(f−1Af)≥1=f−1A≥1f−f−1A≥1(f),

(16)((∂q(f))−1∂q⋅A⋅∂q−1∂q(f))≥1=(∂q(f))−1∂qA≥1∂q−1∂q(f) −(∂q(f))−1(∂qA≥1)(f),

(17)(∂q−1fAf−1∂q)≥1=∂q−1fA≥1f−1∂q−∂q−1f−1A≥1∗(f)∂q,

(18)(Af∂q−1)<0=A≥0(f)∂q−1+A<0f∂q−1,

(19)(∂q−1fA)<0=∂q−1A≥0∗(f)+∂q−1fA<0,

(20)f1∂q−1g1f2∂q−1g2=f1∂q−1(g1f2)∂q−1g2−f1∂q−1θ(∂q−1(g1f2))g2.

Proof. Equations (15) and (16) can be found in [8], (18) and (19) are proved in [7] and (20) can be obtained by direct computation. Here we only prove (17).

(∂q−1fAf−1∂q)≥1=(∂q−1fA≥1f−1)≥0∂q=∂q−1fA≥1f−1∂q−(∂q−1fA≥1f−1)<0∂q=∂q−1fA≥1f−1∂q−∂q−1f−1A≥1∗(f)∂q.

After the preparation above, one can find the following proposition [8], [9] about the q-mKP hierarchy by using Lemma 1 and Lemma 2.

Proposition 3.There are three elementary gauge transformation operators for the q-mKP hierarchy, i.e.

(21)T1(Φ)=Φ−1,

(22)T2(Φ)=(∂q(Φ))−1∂q,

(23)T3(Ψ)=∂q−1Ψ.

where Φ≠0 and Ψ are the eigenfunction and the adjoint eigenfunction of the q-mKP hierarchy [see (11)], respectively. In particular, Φ is not a constant in T₂.

Remark 1. The T₃ here is different from that in [8]. This one is generated by the adjoint eigenfunction.

Further, one can obtain the proposition below [8], [9].

Proposition 4.Under the gauge transformation operator T₁(Φ), T₂(Φ) and T₃(Ψ), the change of Z, Φ₁, Ψ₁is showed in Table 1.

Table 1:

Elementary gauge transformations q-mKP→q-mKP.

Lq-mKP→Lq-mKP(1)	Z⁽¹⁾=	Φ1(1)=	Ψ1(1)=
T₁=Φ⁻¹	Φ⁻¹Z	Φ⁻¹Φ₁	ΦΨ₁
T₂=(∂_q(Φ))⁻¹∂_q	(∂q(Φ))−1∂qZ∂q−1	(∂_q(Φ))⁻¹∂_q(Φ₁)	∂q(Φ)θ∂q−1(Ψ1)
T3=∂q−1Ψ	∂q−1ΨZ∂q	∂q−1(ΨΦ1)	∂_qθ⁻¹(Ψ₁/Ψ)

Φ₁ and Ψ₁ are the eigenfunction and the adjoint eigenfunction of L, respectively. Φ₁≠cΦ and Ψ₁≠cΨ, with c as some constant.

Assume L be the Lax operator of the q-mKP hierarchy, Φ₁ and Φ₂ be two nonzero independent eigenfunctions and Ψ₁ and Ψ₂ be two independent adjoint eigenfunctions. One can consider the following Bianchi diagram (Fig. 1)

Figure 1:

The Bianchi diagram for T_i, i=1, 2, 3.

where i=1, 2, 3 and

(24)Ai={Φ, i=1, 2Ψ, i=3,L^=Ti(A1)LTi(A1)−1, Φ^2=Ti(A1)(Φ2), Ψ^2=(Ti∗(A1))−1(Ψ2),L¯=Ti(A2)LTi(A2)−1, Φ¯1=Ti(A2)(Φ1), Ψ¯1=(Ti∗(A2))−1(Ψ1),L¯^=Ti(A¯1)L¯Ti(A¯1)−1, L^¯=Ti(A^2)L^Ti(A^2)−1.

By direct calculation, we find that the Bianchi diagram cannot commute with themselves and thus they are not convenient in the application. Further, according to Proposition 4, one can also find that T_i with i=1, 2, 3 can not commute with each other, that is

(25)TiTj≠TjTi, i, j=1, 2, 3.

Therefore the gauge transformation operators T_i with i=1, 2, 3 are not applicable in the q-mKP hierarchy. Thus we must seek other kinds of gauge transformation operators which can commute with each other in the q-mKP hierarchy, so we introduce T_D(Φ) and T_I(Ψ) next.

As 1 is the eigenfunction of the q-mKP hierarchy [see (11)], one can define [8], [9]

(26)TD(Φ)≜T2(1(1))T1(Φ)=(∂q(Φ−1))−1∂qΦ−1,

(27)TI(Ψ)≜T1(1(1))T3(Ψ)=(∂q−1(Ψ))−1∂q−1Ψ.

Then by the direct computation similar to Proposition 4, one can obtain the next proposition.

Proposition 5.Under the gauge transformation operator T_D (Φ) and T_I(Ψ), the objects in the q-mKP hierarchy are changed in the way shown in Table 2.

Table 2:

Gauge transformations T_D(Φ) and T_I(Ψ).

Lq-mKP→Lq-mKP(1)	Z⁽¹⁾=	Φ1(1)=	Ψ1(1)=
T_D(Φ)	TD(Φ)Z∂q−1	∂_q(Φ₁/Φ)/∂_q(Φ⁻¹)	∂q(Φ−1)θ∂q−1(ΦΨ1)
T_I(Ψ)	T_I(Ψ)Z∂_q	∂q−1(ΨΦ1)(∂q−1(Ψ))−1	∂q−1(Ψ)∂qθ−1(Ψ−1Ψ1)

According to Table 2, one finds that

(28)TD(Φ)(Φ)=0, (TI(Ψ)−1)∗(Ψ)=0,

(29)TD(Φ)(1)=1, TI(Ψ)(1)=1.

Lemma 6.T_D(Φ) and T_I(Ψ) commute with each other, i.e

(30)TD(Φ2(1))TD(Φ1)=TD(Φ1(1))TD(Φ2),

(31)TD(Φ(1))TI(Ψ)=TI(Ψ(1))TD(Φ),

(32)TI(Ψ2(1))TI(Ψ1)=TI(Ψ1(1))TI(Ψ2)

where “(1)” means the objects changed under the corresponding right gauge transformation operator, according to the rules in Table 2.

Proof. As (30) was proved in [8], here we will mainly prove (31) and (32) next.

As for (31)

(33)TD(Φ(1))TI(Ψ) =(∂q((Φ(1))−1))−1⋅θ((Φ(1))−1(∂q−1(Ψ))−1)⋅Ψ +(∂q((Φ(1))−1))−1⋅∂q((Φ(1))−1(∂q−1(Ψ))−1)⋅∂q−1⋅Ψ =1∂q((Φ(1))−1)(θ((∂q−1(ΨΦ))−1)⋅Ψ+∂q((∂q−1(ΨΦ))−1)⋅∂q−1⋅Ψ) =1∂q−1(ΨΦ)−Φ∂q−1(Ψ)(∂q−1(ΨΦ)−Φ∂q−1Ψ)

where we have used the following relations

∂q(gf)=(θ(f)∂q+∂q(f))(g),∂q(f−1)=−θ(f−1)∂q(f)⋅f−1

and

(34)TI(Ψ(1))TD(Φ)=(∂q−1(Ψ(1)))−1∂q−1⋅Ψ(1)⋅(∂q(Φ−1))−1⋅∂q⋅Φ−1 =(∂q−1(Ψ(1)))−1(∂q−1(ΨΦ)⋅Φ−1−∂q−1⋅Ψ) =(∂q−1(∂q(Φ−1)θ∂q−1(ΨΦ)))−1(∂q−1(ΨΦ)⋅Φ−1−∂q−1⋅Ψ) =1∂q−1(ΨΦ)−Φ∂q−1(Ψ)(∂q−1(ΨΦ)−Φ∂q−1Ψ).

By comparing (33) and (34), one can find (31) is right.

As for (32), using (20)

(35)TI(Ψ2(1))TI(Ψ1) =(∂q−1(Ψ2(1)))−1∂q−1Ψ2(1)(∂q−1(Ψ1))−1∂q−1Ψ1 =1∂q−1(Ψ2(1))(∂q−1(Ψ2(1)(∂q−1(Ψ1))−1)⋅∂q−1⋅Ψ1 −∂q−1⋅θ(∂q−1(Ψ2(1)(∂q−1(Ψ1))−1))Ψ1) =1θ−1(Ψ1−1Ψ2)∂q−1(Ψ1)−∂q−1(Ψ2) (θ−1(Ψ1−1Ψ2)∂q−1⋅Ψ1−∂q−1⋅Ψ2) =1θ−1(Ψ1)∂q−1(Ψ2)−θ−1(Ψ2)∂q−1(Ψ1) (θ−1(Ψ1)∂q−1⋅Ψ2−θ−1(Ψ2)∂q−1⋅Ψ1).

By exchanging Ψ₁ and Ψ₂, one can obtain (32).

Remark 2. From this proposition, T_D(Φ) and T_I(Ψ) are more applicable in the case of the q-mKP hierarchy. Therefore, we would mainly discuss T_D(Φ) and T_I(Ψ) next.

4 Gauge Transformation Operators T_D(Φ) and T_I(Ψ)

Consider the following chain of the gauge transformation operators T_D(Φ) and T_I(Ψ)

L→TD(Φ1)L(1)→TD(Φ2(1))L(1)→⋯→L(n−1)→TD(Φn(n−1))L(n)→TI(Ψ1(n))L(n+1)→TI(Ψ2(n+1))⋯→L(n+k−1)→TI(Ψk(n+k−1))L(n+k).

Denote

(36)T(n,k)=TI(Ψk(n+k−1))⋯TI(Ψ1(n))TD(Φn(n−1))⋯TD(Φ2(1))TD(Φ1).

where Φi(l) means the action of l-term T_D or T_I on Φ_i. Ψi(l) means the inverse and adjoint action of l-term T_D or T_I on Ψ.

Next, we will compute the explicit form of T^(n,k) in terms of Φ_i and Ψ_i. Before this, the following lemma is needed.

Lemma 7.T^(0,k)and T^(n,0)have the forms below

(37)T(0,k)=∑i=1kαi∂q−1⋅Ψi,

(38)(T(n,0))−1=∑i=1nΦi∂q−1⋅βi.

Proof. Here we only prove the first identity, as the second one is similar. Assume this lemma holds for k−1, then according to (20) and Table 2

(39)T(0,k)=∑i=2kαi∂q−1Ψi(1)(∂q−1(Ψ1))−1∂q−1Ψ1=∑i=2kαi∂q−1(Ψi(1)(∂q−1(Ψ1))−1)∂q−1Ψ1−∑i=2kαi∂q−1θ∂q−1(Ψi(1)(∂q−1(Ψ1))−1)Ψ1=∑i=2k(αiθ−1(Ψ1−1Ψi)∂q−1Ψ1−αi∂q−1⋅Ψi),

by noting that

(40)∂q−1(Ψi(1)(∂q−1(Ψ1))−1)Ψ1=θ−1(Ψ1−1Ψi).

so (37) still holds for k.

The generalised q-Wronskian determinant [3] is defined in the form below, and the following three propositions are related to this.

IWk,nq(Ψk, ⋯, Ψ1;Φ1, ⋯, Φn)=|∂q−1(Φ1Ψk)∂q−1(Φ2Ψk)⋯∂q−1(ΦnΨk)∂q−1(Φ1Ψk−1)∂q−1(Φ2Ψk−1)⋯∂q−1(ΦnΨk−1)⋮⋮⋮⋮∂q−1(Φ1Ψ1)∂q−1(Φ2Ψ1)⋯∂q−1(ΦnΨ1)Φ1Φ2⋯Φn∂q(Φ1)∂q(Φ2)⋯∂q(Φn)⋮⋮⋮⋮∂qn−k−1(Φ1)∂qn−k−1(Φ2)⋯∂qn−k−1(Φn)|

When k=0, the generalised q-Wronskian is reduced to

Wnq(Φ1, ⋯, Φn)=|Φ1Φ2⋯Φn∂q(Φ1)∂q(Φ2)⋯∂q(Φn)⋮⋮⋮⋮∂qn−1(Φ1)∂qn−1(Φ2)⋯∂qn−1(Φn)|.

Proposition 8.When n>k, T^(n,k)and (T^(n,k))⁻¹have the forms below

(41)T(n,k)=1IWk,n+1q(Ψk, Ψk−1, ⋯, Ψ1;Φ1, ⋯, Φn, 1)|∂q−1(Φ1Ψk)∂q−1(Φ2Ψk)⋯∂q−1(ΦnΨk)∂q−1Ψk∂q−1(Φ1Ψk−1)∂q−1(Φ2Ψk−1)⋯∂q−1(ΦnΨk−1)∂q−1Ψk−1⋮⋮⋮⋮⋮∂q−1(Φ1Ψ1)∂q−1(Φ2Ψ1)⋯∂q−1(ΦnΨ1)∂q−1Ψ1Φ1Φ2⋯Φn1∂q(Φ1)∂q(Φ2)⋯∂q(Φn)∂q⋮⋮⋮⋮⋮∂qn−k(Φ1)∂qn−k(Φ2)⋯∂qn−k(Φn)∂qn−k|

and

(42)(T(n,k))−1=(−1)n−1q−kIWk,n+1q(Ψk, Ψk−1, ⋯, Ψ1;Φ1, ⋯, Φn, 1)IWk,nq(Ψk, Ψk−1, ⋯, Ψ1;Φ1, ⋯, Φn)θ(IWk,nq(Ψk, Ψk−1, ⋯, Ψ1;Φ1, ⋯, Φn))|Φ1∂q−1∂q−1θ(ΨkΦ1)⋯∂q−1θ(Ψ1Φ1)θ(Φ1)θ(∂q(Φ1))⋯θ(∂qn−k−2(Φ1))Φ2∂q−1∂q−1θ(ΨkΦ2)⋯∂q−1θ(Ψ1Φ2)θ(Φ2)θ(∂q(Φ2))⋯θ(∂qn−k−2(Φ2))⋮⋮⋮⋮⋮⋮⋮⋮Φn∂q−1∂q−1θ(ΨkΦn)⋯∂q−1θ(Ψ1Φn)θ(Φn)θ(∂q(Φn))⋯θ(∂qn−k−2(Φn))|

Here the determinant of T^(n,k)is expanded by the last column and functions are placed before operators when computing (T^(n,k))⁻¹. The determinant of (T^(n,k))⁻¹is expanded by the first column and functions are on the right hand side. And also the coefficient function before the determinant should be placed after the operatorsΦj∂q−1.

Proof. When n>k, according to (36) and the commutativity of T_D and T_I given in Lemma 6, one can rewrite T^(n,k) as

(43)T(n,k)=AT(0,k)=B−1T(n,0),

where

(44)A=TD(Φn(n+k−1))⋯TD(Φ2(k+1))TD(Φ1(k)),

(45)B=(TI(Ψk(n+k−1))TI(Ψk−1(n+k−2))⋯TI(Ψ1(n)))−1.

Then one can use Lemma 7, and also the fact (T^(n,k))⁻¹=((T^(n,k))⁻¹)_,

(46)(T(n,k))−=(AT(0,k))−=(A∑i=1kαi∂q−1Ψi)−=∑i=1kA(αi)∂q−1Ψi,

(47)(T(n,k))−1=((T(n,0))−1B)−=(∑i=1nΦi∂q−1βiB)−=∑i=1nΦi∂q−1B∗(βi).

Thus T^(n,k) and (T^(n,k))⁻¹ have the forms below

(48)T(n,k)=∑i=0n−kai∂qi+∑i=−k−1ai∂q−1Ψ−i,

(49)(T(n,k))−1=∑j=1nΦj∂q−1bj,

where a_i and b_j are the functions that will be determined below.

Then according to Table 2, one can find

(50)T(n,k)(Φi)=0, T(n,k)(1)=1, i=1, 2, ⋯, n,

that is

(51){∑i=0n−kai∂qi(Φj)+∑i=−k−1ai∂q−1(Ψ−iΦj)=0, j=1, 2, ⋯, n,a0+∑i=−k−1ai∂q−1(Ψ−i)=1,

where (48) is used. By solving this (41) can be obtained.

On the other hand, (48), (49) and Table 2 can lead to

(52)((T(n,k))−1)∗=∑l=0+∞(−1)l+1ql2+l+22∑j=1nbjθ1+l(∂1ql(Φj))∂1q−1−l,

(53)((T(n,k))−1)∗=(−1)k−nqn−kan−k−1∂1qk−n+(lower order terms),

(54)((T(n,k))−1)∗=−∑j=1nbjq∂1q−1Φj,

(55)((T(n,k))−1)∗(Ψj)=0,

Here we have used ∂q∗=−1q∂1q, thus

(56){∑i=1nbiθj+1(∂1qj(Φi))=0, j=0, 1, 2, ⋯, n−k−2,∑i=1nbiθn−k(∂1qn−k−1(Φi))=q(n−k−1)(−n+k+2)2an−k−1=q(n−k−1)(−n+k+2)2IWk,n+1q(Ψk, Ψk−1, ⋯, Ψ1;Φ1, ⋯, Φn, 1)IWk,nq(Ψk, Ψk−1, ⋯, Ψ1;Φ1, ⋯, Φn),∑i=1nbi∂1q−1(ΦiΨj)=0, j=1, 2, ⋯, k,

By solving (56) and with the help of

(57)∂1ql=q−l(l−1)2θ−l∂ql, l=1, 2, 3, ⋯

(42) is obtained.

Similar to the case of n>k,T^(n,n) and (T^(n,k))⁻¹ have the forms below

(58)T(n,n)=a0+∑i=−n−1ai∂q−1Ψ−i,

(59)(T(n,n))−1=a0−1+∑j=1nΦj∂q−1bj,

and one can achieve the coming proposition.

Proposition 9.When n=k,T^(n,k)and (T^(n,k))⁻¹have the following forms

(60)T(n,n)=1IWn,n+1q(Ψn, Ψn−1, ⋯, Ψ1;Φ1, ⋯, Φn, 1)|∂q−1(Φ1Ψn)∂q−1(Φ2Ψn)⋯∂q−1(ΦnΨn)∂q−1Ψn∂q−1(Φ1Ψn−1)∂q−1(Φ2Ψn−1)⋯∂q−1(ΦnΨn−1)∂q−1Ψn−1⋮⋮⋮⋮⋮∂q−1(Φ1Ψ1)∂q−1(Φ2Ψ1)⋯∂q−1(ΦnΨ1)∂q−1Ψ1Φ1Φ2⋯Φn1|

and

(61)(T(n,n))−1=−q−nIWn,n+1q(Ψn, Ψn−1, ⋯, Ψ1;Φ1, ⋯, Φn, 1)IWn,nq(Ψn, Ψn−1, ⋯, Ψ1;Φ1, ⋯, Φn)θ(IWn,nq(Ψn, Ψn−1, ⋯, Ψ1;Φ1, ⋯, Φn))|−1q−1Ψnq−1Ψn−1⋯q−1Ψ1Φ1∂q−1∂q−1θ(ΨnΦ1)∂q−1θ(Ψn−1Φ1)⋯∂q−1θ(Ψ1Φ1)Φ2∂q−1∂q−1θ(ΨnΦ2)∂q−1θ(Ψn−1Φ2)⋯∂q−1θ(Ψ1Φ2)⋮⋮⋮⋮⋮Φn∂q−1∂q−1θ(ΨnΦn)∂q−1θ(Ψn−1Φn)⋯∂q−1θ(Ψ1Φn)|

Here the determinant of T^(n,k)is expanded by the last column and functions are placed before operators, when computing (T^(n,k))⁻¹. The determinant of (T^(n,k))⁻¹is expanded by the first column and functions are on the right hand side. And also the coefficient function before the determinant should be placed after the operatorsΦj∂q−1.

The generalised tilde q-Wronskian determinant is defined in the form below

IW˜n,kq(Φn, ⋯, Φ1;Ψ1, ⋯, Ψk)=|∂q−1(Ψ1Φn)∂q−1(Ψ2Φn)⋯∂q−1(ΨkΦn)⋮⋮⋮⋮∂q−1(Ψ1Φ1)∂q−1(Ψ2Φ1)⋯∂q−1(ΨkΦ1)θ−1(Ψ1)θ−1(Ψ2)⋯θ−1(Ψk)θ−2(∂q(Ψ1))θ−2(∂q(Ψ2))⋯θ−2(∂q(Ψk))⋮⋮⋮⋮θn−k(∂qk−n−1(Ψ1))θn−k(∂qk−n−1(Ψ2))⋯θn−k(∂qk−n−1(Ψk))|

In the case of n<k, one has the next proposition.

Proposition 10.When n<k, T^(n,k)and (T^(n,k))⁻¹have the following forms

(62)T(n,k)=(−1)k−1IW˜n+1,kq(1, Φn, ⋯, Φ1;Ψ1, ⋯, Ψk)×|∂q−1(ΦnΨ1)⋯∂q−1(Φ1Ψ1)θ−1(Ψ1)θ−2(∂q(Ψ1))⋯θ−k+n+1(∂qk−n−2(Ψ1))∂q−1Ψ1∂q−1(ΦnΨ2)⋯∂q−1(Φ1Ψ2)θ−1(Ψ2)θ−2(∂q(Ψ2))⋯θ−k+n+1(∂qk−n−2(Ψ2))∂q−1Ψ2⋮⋮⋮⋮⋮⋮⋮∂q−1(ΦnΨk)⋯∂q−1(Φ1Ψk)θ−1(Ψk)θ−2(∂q(Ψk))⋯θ−k+n+1(∂qk−n−2(Ψk))∂q−1Ψk|

and

(63)(T(n,k))−1=q−kIW˜n+1,kq(1, Φn, ⋯, Φ1;Ψ1, ⋯, Ψk)IW˜n,kq(Φn, ⋯, Φ1;Ψ1, ⋯, Ψk)θ(IW˜n,kq(Φn, ⋯, Φ1;Ψ1, ⋯, Ψk))×|−Φn∂q−1∂q−1θ(Ψ1Φn)⋯∂q−1θ(ΨkΦn)−Φn−1∂q−1∂q−1θ(Ψ1Φn−1)⋯∂q−1θ(ΨkΦn−1)⋮⋮⋮⋮−Φ1∂q−1∂q−1θ(Ψ1Φ1)⋯∂q−1θ(ΨkΦ1)1Ψ1⋯Ψk−q∂qθ−1∂q(Ψ1)⋯θ−1∂q(Ψk)⋮⋮⋮⋮(−1)k−nq(k−n)(k−n+1)2∂qk−nθn−k∂qk−n(Ψ1)⋯θn−k∂qk−n(Ψk)|

Here the determinant of T^(n,k)is expanded by the last column and functions are placed before operators when computing (T^(n,k))⁻¹, the determinant of (T^(n,k))⁻¹is expanded by the first column and functions are on the right hand side. And also the coefficient function before the determinant should be placed after the operatorsΦj∂q−1and∂qj.

Proof. Firstly when n<k,T^(n,k) and (T^(n,k))⁻¹ have the following forms

(64)T(n,k)=∑i=1kai∂q−1Ψi,

(65)(T(n,k))−1=∑j=0k−n∂qjbj+∑j=1nΦj∂q−1b−j,

which can be obtained in the same way as the case n>k. Then by using Table 2

(66)T(n,k)(Φi)=0, T(n,k)(1)=1,

(67)T(n,k)=αn−k∂qn−k+(lower order terms)=∑j=0∞(−1)jq−j(j+1)2∑i=1kaiθ−j−1(∂qj(Ψi))∂q−1−j.

Therefore

(68){∑i=1kai∂q−1(ΨiΦj)=0, j=1, 2, ⋯, n,∑i=1kai∂q−1(Ψi)=1,∑i=1kaiθ−j−1(∂qj(Ψi))=0, j=0, 1, ⋯, k−n−2,

and

(69)αn−k=∑i=1kai(−1)k−n−1q(k−n−1)(n−k)2θn−k(∂qk−n−1(Ψi)).

By calculating this (62) can be derived and also

(70)αn−k=(−1)nq(k−n−1)(n−k)2θn−kIW˜n,kq(Φn, Φn−1, ⋯, Φ1;Ψ1, ⋯, Ψk)IW˜n+1,kq(1, Φn, Φn−1, ⋯, Φ1;Ψ1, ⋯, Ψk).

On the other hand, from (67), (70) and Table 2,

(71)((T(n,k))−1)∗(Ψi)=0, i=1, 2, ⋯, k,

(72)bk−n=αn−k−1=(−1)nq(k−n−1)(k−n)2θk−nIW˜n+1,kq(1, Φn, Φn−1, ⋯, Φ1;Ψ1, ⋯, Ψk)IW˜n,kq(Φn, Φn−1, ⋯, Φ1;Ψ1, ⋯, Ψk).

Then

(73){∑i=0k−nbiqi−1∂1qi(Ψi)+∑i=1nb−i∂1q−1(ΦiΨj)=0, j=1, 2, ⋯, k,bk−n=αn−k−1=(−1)nq(k−n−1)(k−n)2θk−nIW˜n+1,kq(1, Φn, Φn−1, ⋯, Φ1;Ψ1, ⋯, Ψk)IW˜n,kq(Φn, Φn−1, ⋯, Φ1;Ψ1, ⋯, Ψk),

which leads to (63).

5 Applications of T^(n,k)

The explicit forms of T^(n,k) (41), (42), (60), (61), (62) and (63) were achieved in the above section. In this part, we will mainly investigate the applications of the above explicit formulas. At first, let us study the application of T^(n,k) on the eigenfunction and adjoint eigenfunction of the q-mKP hierarchy. The corresponding results are summarised in the following proposition.

Proposition 11.Under the successive gauge transformation T^(n,k), the eigenfunction Φ (which is not independent to Φ₁, L, Φ_n in T^(n,k)) and the adjoint eigenfunction Ψ (which is not proportional to Ψ₁, L, Ψ_k in T^(n,k)) of the q-mKP hierarchy will become into

when n>k,
(74)Φ(n+k)=IWk,n+1q(Ψk, Ψk−1, ⋯, Ψ1;Φ1, ⋯, Φn, Φ)IWk,n+1q(Ψk, Ψk−1, ⋯, Ψ1;Φ1, ⋯, Φn, 1),
(75)Ψ(n+k)=(−1)n+1q−2kIWk,n+1q(Ψk, Ψk−1, ⋯, Ψ1;Φ1, ⋯, Φn, 1)IWk,nq(Ψk, Ψk−1, ⋯, Ψ1;Φ1, ⋯, Φn)θ(IWk,nq(Ψk, Ψk−1, ⋯, Ψ1;Φ1, ⋯, Φn))× θ(IWk+1,nq(Ψ, Ψk, ⋯, Ψ1;Φ1, ⋯, Φn)).
when n=k,
(76)Φ(n+n)=IWn,n+1q(Ψn, Ψn−1, ⋯, Ψ1;Φ1, ⋯, Φn, Φ)IWn,n+1q(Ψn, Ψn−1, ⋯, Ψ1;Φ1, ⋯, Φn, 1),
(77)Ψ(n+n)=(−1)n+1q−2nIWn,n+1q(Ψn, Ψn−1, ⋯, Ψ1;Φ1, ⋯, Φn, 1)IWn,nq(Ψn, Ψn−1, ⋯, Ψ1;Φ1, ⋯, Φn)θ(IWn,nq(Ψn, Ψn−1, ⋯, Ψ1;Φ1, ⋯, Φn)× θ(IWn,n+1(Φ1, ⋯, Φn;Ψ, Ψn, ⋯, Ψ1)).
when n<k,
(78)Φ(n+k)=IW˜n+1,kq(Φ, Φn, ⋯, Φ1;Ψ1, ⋯, Ψk)IW˜n+1,kq(1, Φn, ⋯, Φ1;Ψ1, ⋯, Ψk),
(79)Ψ(n+k)=q−kIW˜n+1,kq(1, Φn, Φn−1, ⋯, Φ1;Ψ1, ⋯, Ψk)IW˜n,kq(Φn, Φn−1, ⋯, Φ1;Ψ1, ⋯, Ψk)θ(IW˜n,kq(Φn, Φn−1, ⋯, Φ1;Ψ1, ⋯, Ψk))|θ∂q−1(ΨΦn)∂q−1θ(Ψ1Φn)⋯∂q−1θ(ΨkΦn)θ∂q−1(ΨΦn−1)∂q−1θ(Ψ1Φn−1)⋯∂q−1θ(ΨkΦn−1)⋮⋮⋮⋮θ∂q−1(ΨΦ1)∂q−1θ(Ψ1Φ1)⋯∂q−1θ(ΨkΦ1)ΨΨ1⋯Ψk−q∂q(Ψ)θ−1∂q(Ψ1)⋯θ−1∂q(Ψk)⋮⋮⋮⋮(−1)k−nq(k−n)(k−n+1)2∂qk−n(Ψ)θn−k∂qk−n(Ψ1)⋯θn−k∂qk−n(Ψk)|.

Proof. After substituting (41), (42), (60), (61), (62) and (63) into the following expressions,

(80)Φ(n+k)=T(n,k)(Φ), Ψ(n+k)=−((T(n,k))−1)∗(Ψ),

these results can be obtained.

For the zero solution of the q-mKP hierarchy,

(81)Lq-mkp(1)=∂q,

the eigenfunction Φ and the adjoint eigenfunction Ψ of the q-mKP hierarchy are defined as the following forms [3],

(82)∂Φ∂tn=∂qn(Φ), ∂Ψ∂tn=−(∂qn)∗(Ψ),

which possess set of solution {Φ_i, Ψ_i} as follows

(83)Φi(x;t¯)=eq(λi1x)e∑j=1∞tjλi1j+aieq(μi1x)e∑j=1∞tjμ1j,

(84)Ψi(x;t¯)=e1q(−λi2qx)e−∑j=1∞tjλi2j+bie1q(−μi2qx)e−∑j=1∞tjμi2j.

Proposition 12.Starting from the zero solution

(85)Lq−mkp(0)=∂q,

the solution of (7) is achieved.

When n>k
(86)v1(n+k)=IWk,nq(Ψk, Ψk−1, ⋯, Ψ1;Φ1, ⋯, Φn)IWk,n+1q(Ψk, Ψk−1, ⋯, Ψ1;Φ1, ⋯, Φn, 1)θ(IWk,n+1q(Ψk, Ψk−1, ⋯, Ψ1;Φ1, ⋯, Φn, 1)IWk,nq(Ψk, Ψk−1, ⋯, Ψ1;Φ1, ⋯, Φn)),
when n=k
(87)v1(n+k)=IWn,nq(Ψn, Ψn−1, ⋯, Ψ1;Φ1, ⋯, Φn)IWn,n+1q(Ψn, Ψn−1, ⋯, Ψ1;Φ1, ⋯, Φn, 1)θ(IWn,n+1q(Ψn,Ψn−1,⋯,Ψ1;Φ1,⋯,Φn,1)IWn,nq(Ψn,Ψn−1,⋯,Ψ1;Φ1,⋯,Φn)),
when n<k
(88)v1(n+k)=IW˜n,kq(Φn, ⋯, Φ1;Ψ1, ⋯, Ψk)IW˜n+1,kq(1, Φn, ⋯, Φ1;Ψ1, ⋯, Ψk)θ(IW˜n+1,kq(1, Φn, ⋯, Φ1;Ψ1, ⋯, Ψk)IW˜n,kq(Φn, ⋯, Φ1;Ψ1, ⋯, Ψk)),

where Φ_i and Ψ_i are given by (83) and (84).

Proof. Here we only give the proof for the case of n>k, because the proof of other cases are almost the same. Starting from L⁽⁰⁾=∂_q, and

(89)L(n+k)=T(n,k)L(1)(T(n,k))−1=(αn−k∂qn−k+⋯)⋅∂q⋅(βn−k∂qk−n+⋯)

By calculating the highest order term,

(90)αn−k∂qn−k⋅∂q⋅βn−k∂qk−n=IWk,nq(Ψk, Ψk−1, ⋯, Ψ1;Φ1, ⋯, Φn)IWk,n+1q(Ψk, Ψk−1, ⋯, Ψ1;Φ1, ⋯, Φn, 1)∂qn−k+1×θk−n(IWk,n+1q(Ψk, Ψk−1, ⋯, Ψ1;Φ1, ⋯, Φn, 1)IWk,nq(Ψk, Ψk−1, ⋯, Ψ1;Φ1, ⋯, Φn)∂qk−n =IWk,nq(Ψk, Ψk−1, ⋯, Ψ1;Φ1, ⋯, Φn)IWk,n+1q(Ψk, Ψk−1, ⋯, Ψ1;Φ1, ⋯, Φn, 1)×θ(IWk,n+1q(Ψk, Ψk−1, ⋯, Ψ1;Φ1, ⋯, Φn, 1)IWk,nq(Ψk, Ψk−1, ⋯, Ψ1;Φ1, ⋯, Φn))∂q +lower order terms.

where

(91)∂qi⋅f=θi(f)∂qi+lower order terms, i∈ℤ.

which deduces (86).

Here, we take n=k=1 as an example,

(92)v1(1+1)=∂q−1(Φ1Ψ1)IW1,2q(Ψ1;Φ1, 1)θ(IW1,2q(Ψ1;Φ1, 1)∂q−1(Φ1Ψ1))=∂q−1(Φ1Ψ1)∂q−1(Φ1Ψ1)−Φ1∂q−1(Ψ1)θ∂q−1(Φ1Ψ1)−θ(Φ1)θ∂q−1(Ψ1)θ∂q−1(Φ1Ψ1),

which is a solution of (7).

6 Conclusions and Discussions

In Section 3, three types of the elementary gauge transformation operators T_i with i=1, 2, 3 (see Prop. 3) are discussed. The actions of T_i on the dressing operator and the (adjoint) eigenfunctions are presented in Proposition 4. Then the successive applications of gauge transformation operators T_i are shown. At the end of Section 3, it is proved that T_i cannot commute with each other [see (25)], which indicates that T_i is not convenient to research the q-mKP hierarchy.

Thus T_D and T_I [see (26) and (27)] are introduced in Section 4, where T_D and T_I can commute with each other [see (30)–(32)], so they are more applicable in the q-mKP hierarchy than T_i with i=1, 2, 3. Then much effort is devoted to the successive applications of T_D and T_I in Section 4. And the products of n terms of T_D and k terms of T_I, denoted as T^(n,k), are given in Proposition 8 for n>k, Proposition 9 for n=k and Proposition 10 for n<k.

Then, the application of T^(n,k) on the eigenfunction and adjoint eigenfunction of the q-mKP hierarchy is studied. Under the successive gauge transformation T^(n,k). The solutions of the (7) are summarised in three cases of n>k,n=k, and n<k.

At last, the results in this paper are expected to be generalised to the constrained q-mKP hierarchy. They are also hoped to be useful to study other aspects of the q-mKP hierarchy.

Acknowledgments

This work is supported by the Fundamental Research Funds for the Central Universities (Grant no. 2015QNA43).

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Received: 2017-11-29

Accepted: 2018-01-16

Published Online: 2018-02-15

Published in Print: 2018-03-28

Articles in the same Issue

https://doi.org/10.1515/zna-2017-0433

Keywords for this article

Gauge Transformation; q-mKP Hierarchy; Successive Applications

The Successive Applications of Two Types of Gauge Transformations for the q-Deformed Modified Kadomtsev-Petviashvili Hierarchy

Article

Abstract

1 Introduction

2 The q-mKP Hierarchy

3 Elementary Gauge Transformation

4 Gauge Transformation Operators TD(Φ) and TI(Ψ)

5 Applications of T(n,k)

6 Conclusions and Discussions

Acknowledgments

References

Articles in the same Issue

Articles in the same Issue

Articles in the same Issue

4 Gauge Transformation Operators T_D(Φ) and T_I(Ψ)

5 Applications of T^(n,k)