Journal of Nonlinear Mathematical Physics

Volume 25, Issue 1, February 2018, Pages 66 - 85

The gauge transformation of the modified KP hierarchy

Authors
Jipeng Cheng
Department of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu 221116, P. R. China, Email:chengjp@cumt.edu.cnandchengjp@mail.ustc.edu.cn
Received 9 June 2017, Accepted 19 August 2017, Available Online 6 January 2021.
DOI
10.1080/14029251.2018.1440743How to use a DOI?
Keywords
mKP hierarchy; gauge transformation; tau function
Abstract

In this paper, we firstly investigate the successive applications of three elementary gauge transformation operators Ti with i = 1,2,3 for the mKP hierarchy in Kupershmidt-Kiso version, and find that the gauge transformation operators Ti can not commute with each other. Then two types of gauge transformation operators TD and TI constructed from Ti are proved that they can commute with each other. In particular, TI is introduced for the first time in the literature. And the successive applications of TD and TI in the form of T(n,k), which is the product of n terms of TD and k terms of TI, are derived in three cases for different n and k. At last, the corresponding successive applications of TD and TI on the eigenfunction Φ, the adjoint eigenfunction Ψ and the tau functions τ0 and τ1 are considered.

Copyright
© 2018 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. Introduction

The modified Kadomtsev-Petviashvili (mKP) hierarchy is a system of nonlinear differential equations satisfied by the tau functions introduced in the early 1980s in [14, 15]. There are several versions of the Lax representations for mKP hierarchy. All the definitions are trying to transfer the relationship between KdV and mKdV to the KP situation. The first one is known as the nonstandard integrable system, developed by B. A. Kupershmidt and K. Kiso [1618, 26], which is defined through the modification of the KP hierarchy (“standard case”) tnL=[(Ln)0,L] to tnL=[(Ln)1,L] (see Section 2 for more details). The second one is defined by the factorization of the Lax operator of the KP hierarchy [6, 19]. The last one has the representation in terms of differential-difference equations [8, 31, 32]. In this paper, we only consider the first one, i.e., Kupershmidt-Kiso version, since it is comparatively easier to discuss the gauge transformation in the first version than other ones. The mKP hierarchy in Kupershmidt-Kiso version is the particular case of the coupled modified KP hierarchy [32], which are proved that there are two tau functions τ0 and τ1. These two tau functions bring much difficulty in the study of the mKP hierarchy, compared with the KP case [7] only owning a single tau function.

The gauge transformation [2, 26, 27], also called Darboux transformation, provides a simple way to construct solutions for integrable hierarchies. By now, the gauge transformations of many integrable hierarchies have been studied, for example, the KP and mKP hierarchies [13, 9, 26, 27, 29, 30, 34], the BKP and CKP hierarchies [4, 10, 12, 25], the discrete KP and modified discrete KP hierarchies [13, 20, 21, 24, 28], the q−KP and modified q-KP hierarchies [5, 11, 22, 23, 33] and so on. Here in this paper, we will investigate the gauge transformation of the mKP hierarchy in Kupershmidt-Kiso version. There are three elementary gauge transformation operators Ti with i = 1, 2, 3 (see Section 3 for more details) for the mKP hierarchy [26, 29, 30]. Here we find that Ti with i = 1, 2, 3 can not commute with each other and therefore they are inconvenient in the application. Thus two types of gauge transformation operators TD [26] and TI constructed from Ti (see Section 4 for more details) are introduced, which are proved that they can commute with each other. And in particular, as far as I know, TI is introduced for the first time in the literature. These gauge transformation operators can not only construct various solutions, but also provide the understanding of the inner structures of the mKP hierarchy, for example, the explicit forms of the tau functions for the mKP hierarchy in Kupershmidt-Kiso version. The main content of this paper is devoted to the successive applications of TD and TI.

In the KP case, there are two elementary gauge transformation operators, denoted by Td and Ti. As for the successive applications of n terms of Td and k terms of Ti, it is enough to only discuss two cases of n > k and n = k in the KP hierarchy [9], since the case k > n can be derived from the commutativity of Td and Ti, and also the fact that the inverse of the conjugation of Td is Ti and vice versa. However in the mKP case, the inverse of the conjugation of TD is not equal to TI and vice versa. Therefore in the mKP case, if we denote the product of n terms of TD and k terms of TI as T(n,k), all three cases for different n and k should be considered in the successive applications of TD and TI. Thus the derivation of T(n,k) in the case of k > n can not be avoided, which is one of the major differences from the corresponding results in the KP hierarchy. In addition, we give the detailed derivations of the assumption of the form of T(n,k). But just knowing the assumed form of T(n,k) is not enough in the derivations of explicit form of T(n,k). Actually, besides the similar conditions of TD and TI (see (4.3)) to the KP case, the additional conditions (see (4.4)) are very crucial in this paper, which is another difference from those in the KP case. At last, the corresponding successive applications of TD and TI on the eigenfunction Φ, the adjoint eigenfunction Ψ and the tau functions τ0 and τ1 are considered. And some examples of τ0 and τ1 are given.

This paper is organized in the following way. In Section 2, some basic facts about the mKP hierarchy are introduced. The successive applications and the commutativity of the elementary gauge transformation operators Ti with i = 1, 2, 3 are discussed in Section 3. Then in Section 4, the successive applications of the gauge transformation operators TD(Φ) and TI(Ψ) are investigated in three cases for different n and k. Further the application on the (adjoint) eigenfunctions and the tau functions are obtained in Section 5. At last in Section 6, some conclusions and discussions are given.

2. the mKP hierarchy

The mKP hierarchy in Kupershmidt-Kiso version [1618, 26] is defined as the following Lax equation

tnL=[(Ln)1,L],n=1,2,3, (2.1)
with the Lax operator L given by the pseudo-differential operator below
L=+u0+u11+u22+u33+, (2.2)

Here ∂ = ∂x and ui = ui(t1 = x, t2, ⋯). The algebraic multiplication of ∂i with the multiplication operator f is given by the usual Leibnitz rule

if=j0(ij)f(j)ij,i, (2.3)
where f(j)=jfxj . For A=iaii,Ak=ikaii and A<k=i<kaii . The name of the mKP hierarchy comes from the fact that (2.1) contains the mKP equation
4utx=(uxxx6u2ux)x+3uyy+6uxuy+6uxxuydx. (2.4)

In this paper, for any (pseudo-) differential operator A and a function f, the symbol A(f) will indicate the action of A on f, whereas the symbol Af will denote just operator product of A and f, and * stands for the conjugate operation: (AB)* = B*A*, ∂* = −∂, f* = f. Note that there is the term u0 in the Lax operator L (see (2.2)) of the mKP hierarchy, which is different from the case of the KP hierarchy.

Similar to the case of the KP hierarchy, the Lax operator L for the mKP hierarchy can be expressed in terms of the dressing operator Z,

LmKP=ZZ1, (2.5)
where Z is given by
Z=z0+z11+z22+(z01exists ). (2.6)

Then the Lax equation (2.1) is equivalent to

tnZ=(LmKPn)0Z=(ZnZ1)0Z. (2.7)

Define the wave and the adjoint wave functions of the mKP hierarchy in the following way:

w(t,λ)=Z(eξ(t,λ))=w^(t,λ)eξ(t,λ), (2.8)
w*(t,λ)=(Z11)*(eξ(t,λ))=w^*(t,λ)λ1eξ(t,λ), (2.9)
with
ξ(t,λ)=xλ+t2λ2+t3λ3+, (2.10)
w^(t,λ)=z0+z1λ1+z2λ2+, (2.11)
w^*(t,λ)=z01+z1*λ1+z2*λ2+. (2.12)

Then w(t, λ) and w*(t, λ) satisfy the bilinear identity [32] below

resλw(t,λ)w*(t,λ)=1, (2.13)
which is equivalent to the mKP hierarchy. Here resλiaiλi=a1 .

It is proved in [32] that there exist two tau functionsa τ1 and τ0 for the mKP hierarchy in Kupershmidt-Kiso version such that

w(t,λ)=τ0(t[λ1])τ1(t)eξ(t,λ), (2.14)
w*(t,λ)=τ1(t+[λ1])τ0(t)λ1eξ(t,λ). (2.15)

By comparing (2.11) with (2.14), one can find

z0=τ0(t)τ1(t),z1z0=xlnτ0(t). (2.16)

The eigenfunction Φ and the adjoint eigenfunction Ψ of the mKP hierarchy are defined in the identities below,

Φtn=(Ln)1(Φ),Ψtn=(Ln)1*(Ψ). (2.17)

3. Elementary Gauge Transformation

In this section, we will investigate the elementary gauge transformation of the mKP hierarchy. For the mKP hierarchy (2.1), suppose T is a pseudo-differential operator, and

L(1)=TLT1, (3.1)
such that
tnL(1)=[(L(1))1n,L(1)] (3.2)
still holds for the transformed Lax operator L(1), then T is called a gauge transformation operator of the mKP hierarchy. According to (3.2), one can easily obtain the lemma below.

Lemma 3.1.

If the pseudo-differential operator T satisfies

(TLnT1)1=T(Ln)1T1+TtnT1, (3.3)
then T is a gauge transformation operator of the mKP hierarchy.

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

Lemma 3.2.

For any pseudo-differential operator A and arbitrary functions f, f1, f2, g, g1 and g2, one has the following operator identities:

(f1Af)1=f1A1ff1A1(f), (3.4)
(fx1A1fx)1=fx1A11fxfx1(A1(f))x, (3.5)
(1gAg1)1=1gA1g11g1A1*(g), (3.6)
(Af1)<0=A0(f)1+A<0f1, (3.7)
(1gA)<0=1A0*(g)+1gA<0, (3.8)
f11g1f21g2=f1g1f2dx1g2f11g1f2dxg2. (3.9)

Proof. (3.4), (3.5), (3.7) and (3.8) can be found in [26, 27]. (3.9) can be obtained by direct computation. Therefore, we only prove (3.6) here. By using (3.8)

(1gAg1)1=(1gA1g1)0=1gA1g1(1gA1g1)<0=1gA1g11g1A1*(g).

After the preparation above, one can find the following proposition [26, 29, 30] about the mKP hierarchy by using Lemma 3.1 and Lemma 3.2.

Proposition 3.1.

There are three elementary gauge transformation operators for the mKP hierarchy, that is,

T1(Φ)=Φ1, (3.10)
T2(Φ)=Φx1, (3.11)
T3(Ψ)=1Ψ, (3.12)
where Φ ≠ 0 and Ψ are the eigenfunction and the adjoint eigenfunction of the mKP hierarchy (see (2.17)) respectively, and in particular Φ is not a constant in T2.

Further, one can obtain the following proposition.

Proposition 3.2.

Under the gauge transformation operator T1(Φ), T2(Φ) and T3(Ψ), the objects in the mKP hierarchy are transformed in the way shown in Table I.

LmKPLmKP(1) Z(1) = Φ1(1)= Ψ1(1)= τ0(1)= τ1(1)=
T1 = Φ− 1 Φ− 1Z Φ− 1Φ1 ΦΨ1 τ0 Φτ1
T2=Φx1 Φr1Z1 Φx1Φ1x ΦxΨ1dx τ1 Φxτ12/τ0
T3=1Ψ 1ΨZ ΨΦ1dx (Ψ1/Ψ)x Ψτ02/τ1 τ0
Table I

Elementary gauge transformations mKP → mKP

where Φ1 and Ψ1 are the different eigenfunction and the adjoint eigenfunction of LmKP from the generating functions Φ and Ψ of the gauge transformation operators.

Proof. Here we only take T2 as an example and the others can be derived in the similar way. The actions of T2 on Z, Φ1 and Ψ1 can be obtained by direct computation. Therefore we mainly focus on the actions on τ0 and τ1. Firstly, from

Z(1)=Φx1Z1=Φx1(z0+z11+)1=Φx1z0+Φx1(z0x+z1)1+, (3.13)
one can get
z0(1)=Φx1z0,z1(1)=Φx1(z0x+z1). (3.14)

Then by (2.16), one can find

xlnτ0(1)=z1(1)z0(1)=xlnz0+z1z0=xlnz0τ0. (3.15)

Therefore according to (2.16) and (3.14),

τ0(1)=τ0z0=τ1,τ1(1)=τ0(1)z0(1)=Φxτ12τ0. (3.16)

Remark: Note that in [29, 30], there is another gauge transformation operator T=1Ψdx , which can be viewed as the composition of T2 and T3, that is, T3(1))T2(Φ) = ∂−1 Ψ(1) Φ∂ = 1Ψdx , with Ψ(1) = T2(Φ)(Ψ).

Denote the successive applications of gauge transformation operators Ti with i = 1, 2, 3 as

T1(n)(Φ1,,Φn)T1(Φn(n1))T1(Φ3(2))T1(Φ2(1))T1(Φ1), (3.17)
T2(n)(Φ1,,Φn)T2(Φn(n1))T2(Φ3(2))T2(Φ2(1))T2(Φ1), (3.18)
T3(n)(Ψ1,,Ψn)T3(Ψn(n1))T3(Ψ3(2))T3(Ψ2(1))T3(Ψ1). (3.19)

Here Φi and Ψi are the eigenfunctions and the adjoint eigenfunctions of the mKP hierarchy which are different from each other, Φk(m)=Ti(m)(Φ1,,Φm)(Φk) and Ψk(m)=Ti(m)(Ψ1,,Ψm)(Ψk) . The results of the successive applications of gauge transformation operators Ti with i = 1, 2, 3 are listed in the proposition below.

Proposition 3.3.

(1)

T1(n)(Φ1,,Φn)=Φn1, (3.20)

(2)

T2(n)(Φ1,,Φn)=1Wn+1(1,Φ1,,Φn)|Φ1xΦn1,xΦ1(i)Φn1(i)iΦ1(n)Φn1(n)n|, (3.21)

(3)

T3(n)(Ψ1,,Ψn)=(1)n+1Wn1(Ψ1,,Ψn1)|Ψ1Ψ1(1)Ψ1(n2)1Ψ1ΨnΨn(1)Ψn(n2)1Ψn|, (3.22)
where WnWn1, Φ2,⋯,Φn) is the Wronskian determinant of Φ1, Φ2,⋯,Φn. And the determinant of T3(n) is expanded by the last column and the functions are on the left-hand side.

Proof. (1)

T1(n)=T1(Φn(n1))T1(Φ3(2))T1(Φ2(1))T1(Φ1)=(Φ1Φ2(1)Φn2(n3)Φn1(n2)Φn(n1))1=(Φ1Φ2(1)Φn2(n3)Φn(n2))1=(Φ1Φ2(1)Φn(n3))1==Φn1, (3.23)
where we have used the following fact Φk(k1)Φj(k)=Φj(k1) , with Φj(k)=T1(Φk(k1))(Φj(k1)) .

(2) Assume T2(n)=i=1naii . Since T2(n)(Φi)=0,i=1,2,,n1 and T2(n)(Φn)=1 , one can obtain the following linear equation group,

{i=1naiΦk(i)=0,k=1,2,n1,i=1naiΦn(i)=1. (3.24)

Thus

ai=(1)i+n1Wn+1(1,Φ1,,Φn)|Φ1xΦn1,xΦ1(i1)Φn1(i1)Φ1(i+1)Φn1(i+1)Φ1(n)Φn1(n)|. (3.25)

The substitution of (3.25) for T2(n)=i=1naii leads to (3.21).

(3) Firstly, we can prove that T3(n) has the form of T3(n)=i=1nbi1Ψi with bn=(1)n+1 by induction. In fact, we can assume T3(n1)(Ψ2(1),,Ψn(1))=i=2nbi1Ψi(1) with bn = (−1)n. Then according to (3.9) and Proposition 3.2, one can find

T3(n)=T3(n1)(Ψ2(1),,Ψn(1))T3(Ψ1)=i=2nbi1Ψi(1)1Ψ1=i=2n(biΨi(1)dx1Ψ1bi1Ψi(1)dxΨ1)=i=2n(biΨi(1)dx1Ψ1bi1Ψi) (3.26)
still holds with the term (−1)n + 1−1Ψn.

Then by comparing the fact T3(n)=αnn+ (lower order terms) with the form of T3(n) given above, one can obtain the following linear equation group,

{i=1nbiΨi(k)=0,k=0,1,2,n2,bn=(1)n+1. (3.27)

Therefore

bi=(1)i+1Wn1(Ψ1,,Ψn1)|Ψ1Ψi1Ψi+1ΨnΨ1xΨi1,xΨi+1,xΨn,xΨ1(n2)Ψi1(n2)Ψi+1(n2)Ψn(n2)|, (3.28)
which leads to (3.22).

Assume L be the Lax operator of the mKP hierarchy, Φ1 and Φ2 be two nonzero independent eigenfunctions, and Ψ1 and Ψ2 be two independent adjoint eigenfunctions. we consider the following diagram

Figure 1.
The Bianchi diagram for Ti, i = 1, 2, 3.

where i = 1, 2, 3 and

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. (3.29)

The question is whether this diagram will commute, i.e.

Ti(2)(Φ1,Φ2)=Ti(2)(Φ2,Φ1),i=1,2,T3(2)(Ψ1,Ψ2)=T3(2)(Ψ2,Ψ1). (3.30)

In fact, according to Proposition 3.3,

T1(2)(Φ1,Φ2)T1(2)(Φ2,Φ1)=Φ21Φ110, (3.31)

T2(2)(Φ1,Φ2)T2(2)(Φ2,Φ1)=1W3(1,Φ1,Φ2)|Φ1x+Φ2xΦ1xx+Φ2xx2|0, (3.32)
T3(2)(Ψ1,Ψ2)T3(2)(Ψ2,Ψ1)=(Ψ2/Ψ1+1)1Ψ1(Ψ1/Ψ2+1)1Ψ20. (3.33)

Therefore, the Bianchi diagrams for Ti, i = 1, 2, 3 does not commute. Further, according to Proposition 3.2, one can also find that

T1(Φ2(1))T2(Φ1)T2(Φ1(1))T1(Φ2), (3.34)
T1(Φ(1))T3(Ψ)T3(Ψ(1))T1(Φ), (3.35)
T2(Φ(1))T3(Ψ)T3(Ψ(1))T2(Φ). (3.36)

So Ti with i = 1, 2, 3 can not commute with each other, that is,

TiTjTjTi,i,j=1,2,3. (3.37)

This fact tells us that the gauge transformation operators Ti with i = 1, 2, 3 are not convenient when solving the mKP hierarchy. Thus we must seek the other kinds of gauge transformation operators which can commute with each other in the mKP hierarchy, that is, TD(Φ) and TI(Ψ).

4. Gauge Transformation operators TD(Φ) and TI(Ψ)

Since 1 is the eigenfunction of the mKP hierarchy (see (2.17)), one can define

TD(Φ)T2(1(1))T1(Φ)=(Φ1)x1Φ1, (4.1)
TI(Ψ)T1(1(1))T3(Ψ)=(Ψdx)11Ψ. (4.2)

Here TD is introduced in [26], while as far as I know TI is presented for the first time in the literature. Then by the direct computation similar to Proposition 3.2, one can obtain the following proposition.

Proposition 4.1.

Under the gauge transformation operator TD(Φ) and TI(Ψ), the objects in the mKP hierarchy are transformed in the way shown in Table II.

LmKPLmKP(1) Z(1) = Φ1(1)= Ψ1(1)= τ0(1)= τ1(1)=
TD(Φ) TD(Φ)Z− 1 (Φ1/Φ)x/(Φ1)x (Φ1)xΦΨ1dx Φτ1 Φxτ12/τ0
T1(Ψ) T1(Ψ)Z ΨΦ1dx/Ψdx (Ψ1/Ψ)xΨdx Ψτ02/τ1 τ0Ψdx
Table II

gauge transformations TD(Φ) and TI(Ψ)

According to Table II, one finds that

TD(Φ)(Φ)=0,(TI(Ψ)1)*(Ψ)=0, (4.3)
TD(Φ)(1)=1,TI(Ψ)(1)=1. (4.4)

It can be proved that TD(Φ) and TI(Ψ) are commute with each other, i.e.,

TD(Φ2(1))TD(Φ1(0))=TD(Φ1(1))TD(Φ2(0)), (4.5)
TD(Φ(1))TI(Ψ(0))=TI(Ψ(1))TD(Φ(0)), (4.6)
TI(Ψ2(1))TI(Ψ1(0))=TI(Ψ1(1))TI(Ψ2(0)). (4.7)

Therefore, TD(Φ) and TI(Ψ) are more applicable in the case of the mKP hierarchy.

Consider the following chain of the gauge transformation operators TD(Φ) and TI(Ψ)

LTD(Φ1)L(1)TD(Φ2(1))L(2)L(n1)TD(Φn(n1))L(n)TI(Ψ1(n))L(n+1)TI(Ψ1(n+1))L(n+k1)TI(Ψk(n+k1))L(n+k).

Denote

T(n,k)=TI(Ψk(n+k1))TI(Ψ1(n))TD(Φn(n1))TD(Φ2(1))TD(Φ1). (4.8)

Next, we will compute the explicit form of T(n,k) in terms of Φi and Ψi. Before this, the lemma below is needed.

Lemma 4.1.

T(0, k) and T(n,0) have the following form

T(0,k)=i=1kαi1Ψi, (4.9)
(T(n,0))1=i=1nΦi1βi. (4.10)

Proof. This lemma can be proved by induction. Here we only prove the first identity, since the second one is almost the same. Assume this lemma holds for k − 1, then according to (3.9) and Table II

T(0,k)=i=2kαi1Ψi(1)(Ψ1dx)11Ψ1=i=2kαi(Ψi(1)(Ψ1dx)1)dx1Ψ1i=2kαi1(Ψi(1)(Ψ1dx)1)dxΨ1=i=2k(αiΨi/Ψ11Ψ1αi1Ψi), (4.11)
by noting that
(Ψi(1)(Ψ1dx)1)dxΨ1=Ψi. (4.12)

The generalized Wronskian determinant [9] is needed in the next propositions, which is defined in the following form

IWk,nIWk,n(Ψk,Ψk1,,Ψ1;Φ1,,Φn)=|Φ1ΨkdxΦ2ΨkdxΦnΨkdxΦ1Ψk1dxΦ2Ψk1dxΦnΨk1dxΦ1Ψ1dxΦ2Ψ1dxΦnΨ1dxΦ1Φ2ΦnΦ1xΦ2xΦnxΦ1(nk1)Φ2(nk1)Φn(nk1)|.

In particular,

IW0,n=Wn(Φ1,Φ2,,Φn), (4.13)
which is Wronskian determinant of Φ1, Φ2,⋯, Φn.

Proposition 4.2.

When n > k, T(n,k) and (T(n,k)) − 1 have the following forms

T(n,k)=1IWk,n+1(Ψk,Ψk1,,Ψ1;Φ1,,Φn,1)|Φ1ΨkdxΦ2ΨkdxΦnΨkdx1ΨkΦ1Ψk1dxΦ2Ψk1dxΦnΨk1dx1Ψk1Φ1Ψ1dxΦ2Ψ1dxΦnΨ1dx1Ψ1Φ1Φ2Φn1Φ1xΦ2xΦnxΦ1(nk)Φ2(nk)Φn(nk)nk| (4.14)
and
(T(n,k))1=(1)n1IWk,n+1(Ψk,Ψk1,,Ψ1;Φ1,,Φn,1)IWk,n(Ψk,Ψk1,,Ψ1;Φ1,,Φn)2|Φ11ΨkΦ1dxΨ1Φ1dxΦ1Φ1(nk2)Φ21ΨkΦ2dxΨ1Φ2dxΦ2Φ2(nk2)Φn1ΨkΦndxΨ1ΦndxΦnΦn(nk2)|. (4.15)

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

Proof. When n > k, according to (4.8) and the commutativity of TD and TI given in (4.5)(4.7), one can rewrite T(n,k) as

T(n,k)=AT(0,k)=B1T(n,0), (4.16)
where
A=TD(Φn(n+k1))TD(Φ2(k+1))TD(Φ1(k)), (4.17)
B=(TI(Ψk(n+k1))TI(Ψk1(n+k2))TI(Ψ1(n)))1. (4.18)

Then by using Lemma 3.2 and Lemma 4.1, and also the fact (T(n,k))1=((T(n,k))1) ,

(T(n,k))=(AT(0,k))=(Ai=1kαi1Ψi)=i=1kA(αi)1Ψi, (4.19)
(T(n,k))1=((T(n,0))1B)=(i=1nΦi1βiB)=i=1nΦi1B*(βi). (4.20)

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

T(n,k)=i=0nkaii+i=k1ai1Ψi, (4.21)
(T(n,k))1=j=1nΦj1bj, (4.22)
where ai and bj are the functions that will be determined below.

Then from Table II, one can find

T(n,k)(Φi)=0,T(n,k)(1)=1,i=1,2,,n, (4.23)
that is
{i=0nkaiΦj(i)+i=k1aiΨiΦjdx=0,j=1,2,,n,a0+i=k1aiΨidx=1, (4.24)
where (4.21) is used. By solving this equation, (4.14) can be proved.

On the other hand, (4.21), (4.22) and Table II can lead to

((T(n,k))1)*=l=0+(1)l+1j=1nbjΦj(l)1l, (4.25)
((T(n,k))1)*=(1)nkank1n+k+(lower order terms), (4.26)
((T(n,k))1)*(Ψj)=0, (4.27)

Thus

{i=1nbiΦiΨjdx=0,j=k,k1,,1,i=1nbiΦi(j)=0,j=0,1,2,,nk2,i=1nbiΦi(nk1)=ank1=IWk,n+1(Ψk,Ψk1,,Ψ1;Φ1,,Φn,1)IWk,n(Ψk,Ψk1,,Ψ1;Φ1,,Φn), (4.28)
which gives rise to (4.15).

By the same manner as the case n > k, one can show that T(n,n) and (T(n,n))−1 have the similar forms

T(n,n)=a0+i=n1ai1Ψi, (4.29)
(T(n,n))1=a01+j=1nΦj1bj, (4.30)
and thus have the following proposition.

Proposition 4.3.

When n = k, T(n,k) and (T(n,k)) − 1 have the following forms

T(n,n)=1IWn,n+1(Ψn,Ψn1,,Ψ1;Φ1,,Φn,1)|Φ1ΨndxΦ2ΨndxΦnΨndx1ΨnΦ1Ψn1dxΦ2Ψn1dxΦnΨn1dx1Ψn1Φ1Ψ1dxΦ2Ψ1dxΦnΨ1dx1Ψ1Φ1Φ2Φn1|, (4.31)
and
(T(n,n))1=IWn,n+1(Ψn,Ψn1,,Ψ1;Φ1,,Φn,1)IWn,n(Ψn,Ψn1,,Ψ1;Φ1,,Φn)2|1ΨnΨn1Ψ1Φ11ΨnΦ1dxΨn1Φ1dxΨ1Φ1dxΦ21ΨnΦ2dxΨn1Φ2dxΨ1Φ2dxΦn1ΨnΦndxΨn1ΦndxΨ1Φndx|. (4.32)

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

As for the case n < k, one has the proposition below.

Proposition 4.4.

When n < k, T(n,k) and (T(n,k)) − 1 have the following forms

T(n,k)=(1)k1IWn+1,k(1,Φn,Φn1,,Φ1;Ψ1,,Ψk)|ΦnΨ1dxΦ1Ψ1dxΨ1Ψ1(kn2)1Ψ1ΦnΨ2dxΦ1Ψ2dxΨ2Ψ2(kn2)1Ψ2ΦnΨkdxΦ1ΨkdxΨkΨk(kn2)1Ψk|, (4.33)
and
(T(n,k))1=IWn+1,k(1,Φn,Φn1,,Φ1;Ψ1,,Ψk)IWn,k(Φn,Φn1,,Φ1;Ψ1,,Ψk)2|Φn1ΦnΨ1dxΦnΨkdxΦ11Φ1Ψ1dxΦ1Ψkdx1Ψ1ΨkΨ1xΨkx()knΨ1(kn)Ψk(kn)|. (4.34)

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

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

T(n,k)=i=1kai1Ψi, (4.35)
(T(n,k))1=j=0knjbj+j=1nΦj1bj, (4.36)
which can be obtained in the same way as the case n>k. Then by using Table II
T(n,k)(Φi)=0,T(n,k)(1)=1, (4.37)
T(n,k)=αnknk+(lower order terms)=j=0(1)ji=1kaiΨi(j)1j. (4.38)

Therefore

{i=1kaiΨidx=1,i=1kaiΨiΦjdx=0,j=n,n1,,1,i=1kaiΨi(j)=0,j=0,1,,kn2. (4.39)

By solving this equation, (4.33) can be derived. And also by using

αnk=i=1k(1)1+knaiΨi(1+kn), (4.40)
one can obtain
αnk=(1)nIWnk(Φn,Φn1,,Φ1;Ψ1,,Ψk)IWn+1,k(1,Φn,Φn1,,Φ1;Ψ1,,Ψk). (4.41)

On the other hand, from (4.38), (4.41) and Table II,

((T(n,k))1)*(Ψi)=0,i=1,2,,k, (4.42)
bkn=αnk1=(1)nIWn+1,k(1,Φn,Φn1,,Φ1;Ψ1,,Ψk)IWn,k(Φn,Φn1,,Φ1;Ψ1,,Ψk). (4.43)

Then

{i=0kn(1)ibiΨj(i)i=1nbiΦiΨjdx=0,j=1,2,,k,bkn=(1)nIWn+1,k(1,Φn,Φn1,,Φ1;Ψ1,,Ψk)IWn,k(Φn,Φn1,,Φ1;Ψ1,,Ψk), (4.44)
which leads to (4.34).

5. Applications of T(n,k)

In the last section, the explicit forms of T(n,k) have been obtained. In this section, we will investigate the applications of the above explicit formulas.

Firstly, let’s consider the applications of T(n,k) on the eigenfunction and adjoint eigenfunction of the mKP hierarchy. We summarize the corresponding results in the next proposition.

Proposition 5.1.

Under the successive gauge transformation T(n,k), the eigenfunction Φ (which is not proportional to Φ1, ⋯, Φn in T(n,k)) and the adjoint eigenfunction Ψ (which is not proportional to Ψ1, ⋯, Ψk in T(n,k)) of the mKP hierarchy will become into

  • when n > k,

    Φ(n+k)=IWk,n+1(Ψk,Ψk1,,Ψ1;Φ1,,Φn,Φ)IWk,n+1(Ψk,Ψk1,,Ψ1;Φ1,,Φn,1) (5.1)
    Ψ(n+k)=(1)n+1IWk,n+1(Ψk,Ψk1,,Ψ1;Φ1,,Φn,1)IWk,n(Ψk,Ψk1,,Ψ1;Φ1,,Φn)2 (5.2)
    ×IWk+1,n(Ψ,Ψk,,Ψ1;Φ1,,Φn) (5.3)

  • when n = k

    Φ(n+n)=IWn,n+1(Ψn,Ψn1,,Ψ1;Φ1,,Φn,Φ)IWn,n+1(Ψn,Ψn1,,Ψ1;Φ1,,Φn,1) (5.4)
    Ψ(n+n)=(1)n+1IWn,n+1(Ψn,Ψn1,,Ψ1;Φ1,,Φn,1)IWn,n(Ψn,Ψn1,,Ψ1;Φ1,,Φn)2×IWn,n+1(Φ1,,Φn;Ψ,Ψn,,Ψ1). (5.5)

  • when n < k

    Φ(n+k)=IWn+1,k(Φ,Φn,,Φ1;Ψ1,,Ψk)IWn+1,k(1,Φn,,Φ1;Ψ1,,Ψk), (5.6)
    Ψ(n,k)=IWn+1,k(1,Φn,Φn1,,Φ1;Ψ1,,Ψk)IWn,k(Φn,Φn1,,Φ1;Ψ1,,Ψk)2×IWn,k+1(Φn,,Φ1;Ψ,Ψ1,,Ψk). (5.7)

    Proof. These results can be obtained by substituting (4.14), (4.15), (4.31), (4.32), (4.33) and (4.34) into the following expressions

    Φ(n+k)=T(n,k)(Φ),Ψ(n+k)=((T(n,k))1)*(Ψ). (5.8)

Proposition 5.2.

Under the successive gauge transformation T(n,k), the tau functions τ0 and τ1 of the mKP hierarchy will be changed into

  • when nk

    τ0(n+k)=IWk,n(Ψk,Ψk1,,Ψ1;Φ1,,Φn)τ1nkτ0nk1, (5.9)
    τ1(n+k)=IWk,n+1(Ψk,Ψk1,,Ψ1;Φ1,,Φn,1)τ1nk+1τ0nk; (5.10)

  • when n < k

    τ0(n+k)=IWn,k(Φn,Φn1,,Φ1;Ψ1,,Ψk)τ1nkτ0nk1, (5.11)
    τ1(n+k)=(1)nIWn+1,k(1,Φn,Φn1,,Φ1;Ψ1,,Ψk)τ1nk+1τ0nk. (5.12)

Proof. Here we only give the proof for the case of nk, since the proof in the other case is almost the same. For this, assume

T(n,k)=αnknk+αnk1nk1+. (5.13)

Then through (4.14) and (4.31), one can find

αnk=IWk,n(Ψk,Ψk1,,Ψ1;Φ1,,Φn)IWk,n+1(Ψk,Ψk1,,Ψ1;Φ1,,Φn,1),
αnk1=x(IWk,n(Ψk,Ψk1,,Ψ1;Φ1,,Φn))IWk,n+1(Ψk,Ψk1,,Ψ1;Φ1,,Φn,1). (5.14)

Further from the transformation

Z(n+k)=T(n,k)Z(0)kn, (5.15)
one can obtain
z0(n+k)=αnkz0,z1(n+k)=αnkz1+(nk)αnkz0x+αnk1z0. (5.16)

Then according to (5.14) and the relation between τ0 and τ1 given in (2.16), one can find

xln(τ0(n+k))=z1(n+k)z0(n+k)=z1z0+(nk)xln(z0)+αnk1αnk=xln(τ0)+(nk)xln(z0)xln(IWk,n(Ψk,Ψk1,,Ψ1;Φ1,,Φn)),

Therefore by (2.16), (5.14) and (5.16),

τ0(n+k)=IWk,n(Ψk,,Ψ1;Φ1,,Φn)τ0z0nk=IWk,n(Ψk,,Ψ1;Φ1,,Φn)τ1nkτ0nk1,τ1(n+k)=τ0(n+k)z0(n+k)=IWk,n+1(Ψk,Ψk1,,Ψ1;Φ1,,Φn,1)τ1nk+1τ0nk.

In order to obtain the explicit examples of τ0 and τ1, we start from the zero solution, i.e., L(0) = ∂, which satisfies (2.1). One can easily find that in this case, the mKP hierarchy in Kupershmidt-Kiso version coincides with the usual KP hierarchy. Denote

Φi(0)=exp(ξ(t,αi))+aiexp(ξ(t,βi)),i=1,2,,n, (5.17)
Ψj(0)=exp(ξ(t,γj))+bjexp(ξ(t,δj)),j=1,2,,k, (5.18)
where αi, βi, γj and δj are distinct complex numbers for different i or j, and ai and bj are constants. Then one can easily check that Φi(0) and Ψj(0) satisfy (2.17), and thus Φi(0) and Ψj(0) are the corresponding (adjoint) eigenfunctions of the mKP hierarchy in Kupershmidt-Kiso version or the usual KP hierarchy with respect to L(0). And in particular, 1 is also the eigenfunction of the KP hierarchy corresponding to L(0). As for the tau functions in this case, one can easily find that
τ0(0)=1,τ1(0)=1. (5.19)

Then according to Proposition 5.2 and [9], one has the following results

Proposition 5.3.

Starting from the zero solution of the mKP hierarchy, i.e., L(0) = ∂, the tau pair (τ0(n +k), τ1(n +k)) of the mKP hierarchy is showed as follows.

  • when nk,

    τ0(n+k)=IWk,n(Ψk(0),Ψk1(0),,Ψ1(0);Φ1(0),,Φn(0)), (5.20)
    τ1(n+k)=IWk,n+1(Ψk(0),Ψk1(0),,Ψ1(0);Φ1(0),,Φn(0),1); (5.21)

  • when n < k

    τ0(n+k)=IWnk(Φn(0),Φn1(0),,Φ1(0);Ψ1(0),,Ψk), (5.22)
    τ1(n+k)=(1)nIWn+1,k(1,Φn,Φn1,,Φ1;Ψ1,,Ψk). (5.23)

Thus τ0(n+k) and τ1(n+k) can be viewed as the tau functions of the KP hierarchy.

Corollary 5.1.

Starting from L(0) = ∂, the tau pair (τ0(n),τ1(n)) of the mKP hierarchy is listed below,

τ0(n)=Wn(Φ1(0),,Φn(0)), (5.24)
τ1(n)=Wn+1(Φ1(0),,Φn(0),1); (5.25)
which are also the tau functions of the KP hierarchy.

Remark: the above results can help us to understand the tau pair (τ0, τ1) of the mKP hierarchy in Kupershmidt-Kiso version.

6. Conclusions and Discussions

The elementary gauge transformation operators Ti with i = 1, 2, 3 (see Proposition 3.1) are discussed in Section 2. The actions of Ti on the dressing operator, the (adjoint) eigenfunctions and the tau functions are presented in Proposition 3.2. Then the successive applications of gauge transformation operators Ti are shown in Proposition 3.3. At the end of Section 2, Ti are proved that they can not commute with each other (see (3.37)), which reflects that Ti are not easy to carry out for the mKP hierarchy.

Therefore TD and TI (see (4.1) and (4.2)) are introduced in Section 3. And in particular, as far as I know, TI is introduced for the first time in the literature. After that, how the objects of the mKP hierarchy are transformed under TD and TI are listed in Proposition 4.1. And further based upon the results in Proposition 4.1, TD and TI are shown that they can commute with each other (see (4.5)(4.7)) and hence they are more applicable in the mKP hierarchy than Ti with i = 1, 2, 3. Then the main part of Section 3 is devoted to the successive applications of TD and TI. And the products of n terms of TD and k-terms of TI, denoted as T(n,k), are given in Proposition 4.2 for n > k, Proposition 4.3 for n = k and Proposition 4.4 for n < k. At last, the actions of T(n,k) on the eigenfunction Φ, the adjoint eigenfunction Ψ and the tau functions τ0 and τ1 are presented in Proposition 5.1 and 5.2 Also the explicit forms of tau functions τ0 and τ1 are given in Proposition 5.3 and Corollary 5.1.

Compared with the KP hierarchy, there are several differences in the mKP hierarchy, which brings much difficulty to the study of the mKP hierarchy. 1) There is the second high order term in the Lax operator of the mKP hierarchy (see u0 in (2.2)), which leads to the extra condition (see (4.4)) to determine T(n,k). 2) The inverse of the conjugation of TD is not TI and vice versa. Therefore, the case k > n must be discussed in the derivation of T(n,k). In fact, it is more difficult in the case k>n than the cases n>k and n = k, because it is very hard to determine the form of T(n,k) in the case k > n. 3) There are two tau functions τ0 and τ1 in the mKP hierarchy. Usually, it is more inconvenient to study the integrable systems with two tau functions than those with only one single tau function. Hence, additional methods must be developed in the discussion of the tau functions under the successive applications of TD and TI for the mKP hierarchy.

At last, the results in this paper are hoped to be generalized to the constrained mKP hierarchy. Also these results may be helpful to discuss the inner integrability of the mKP hierarchy such as the additional symmetries and the squared eigenfunction symmetries.

Acknowledgments

This work is supported by the Fundamental Research Funds for the Central Universities (Grant No. 2015 QNA 43).

Footnotes

a

Please note that the mKP hierarchy that we consider is the Kupershmidt-Kiso version, which is different from from those in L. A. Dickey’s work about mKP hierarchy [8]. The existence of τ1 and τ0 makes the mKP hierarchy in Kupershmidt-Kiso version become a relatively separate system, just like the KP hierarchy.

References

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[11]JS He, YH Li, and Y Cheng, q-deformed KP hierarchy and q-Deformed constrained KP hierarchy, SIGMA, Vol. 2, 2006, pp. 060.
[25]JJC Nimmo, Darboux transformations from reductions of the KP hierarchy, VG Makhankov, AR Bishop, and DD Holm (editors), Nonlinear Evolution Equations and Dynamical Systems, World Scientific, Singapore, 1995, pp. 168-177.
[28]W Oevel, Darboux transformations for integrable lattice systems, E Alfinito, L Martina, and F Pempinelli (editors), Nonlinear Physics: theory and experiment, World Scientific, Singapore, 1996, pp. 233-240.
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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
25 - 1
Pages
66 - 85
Publication Date
2021/01/06
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1080/14029251.2018.1440743How to use a DOI?
Copyright
© 2018 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Jipeng Cheng
PY  - 2021
DA  - 2021/01/06
TI  - The gauge transformation of the modified KP hierarchy
JO  - Journal of Nonlinear Mathematical Physics
SP  - 66
EP  - 85
VL  - 25
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2018.1440743
DO  - 10.1080/14029251.2018.1440743
ID  - Cheng2021
ER  -