Journal of Nonlinear Mathematical Physics

Volume 26, Issue 1, December 2018, Pages 54 - 68

Solving the constrained modified KP hierarchy by gauge transformations

Authors
Huizhan Chen, Lumin Geng, Na Li, Jipeng Cheng*
School of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu 221116, P. R. China,chengjp@cumt.edu.cn
*Corresponding author.
Corresponding Author
Jipeng Cheng
Received 17 February 2018, Accepted 14 June 2018, Available Online 6 January 2021.
DOI
10.1080/14029251.2019.1544787How to use a DOI?
Keywords
The constrained mKP hierarchy; gauge transformations; successive applications
Abstract

In this paper, we mainly investigate two kinds of gauge transformations for the constrained modified KP hierarchy in Kupershmidt-Kiso version. The corresponding gauge transformations are required to keep not only the Lax equation but also the Lax operator. For this, by selecting the special generating eigenfunction and adjoint eigenfunction, the elementary gauge transformation operators of modified KP hierarchy TD(Φ)=(Φ1)x1Φ1 and TI(Ψ) = Ψ−1 −1Ψx, become the ones in the constrained case. Finally, the corresponding successive applications of TD and TI on the eigenfunction Φ and the adjoint eigenfunction Ψ are discussed.

Copyright
© 2019 The Authors. Published by Atlantis 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

In the early 1980s, the modified Kadomtsev-Petviashvili (mKP) hierarchy [15, 17] is introduced as the nonlinear differential equations satisfied by the tau functions. There are many versions of the mKP hierarchy [6,8,10,16,1820,24,28,29], and their common features are to convert the relationship between KdV and mKdV into the KP situation. Here, we will only consider the Kupershmidt-Kiso version [6, 1820, 24]. The mKP hierarchy in Kupershmidt-Kiso version is the particular case of the coupled modified KP hierarchy [29], which is proved that there are two tau functions τ0 and τ1. Compared with the KP hierarchy [9, 11] only owning a single tau function, it is more difficult to study the mKP hierarchy with two tau functions. The existence of τ1 and τ0 makes the mKP hierarchy in Kupershmidt-Kiso version becomes a relatively separate system, just like the KP hierarchy [9, 11].

Gauge transformation [37, 1214, 2125, 27] is one kind of powerful methods to construct the solutions of the integrable systems. By now, the gauge transformations of many integrable hierarchies have been studied. For example, the KP and mKP hierarchies [3, 4, 6, 13, 24, 25], the BKP and CKP hierarchies [14, 27], the discrete KP and modified discrete KP hierarchies [22, 23], the q-KP and modified q-KP hierarchies [5, 7, 12, 21] and so on. For the mKP hierarchy, there are two types of elementary gauge transformation operators [6, 24]: differential type TD(Φ)=(Φ1)x1Φ1 and integral type TI(Ψ) = Ψ−1−1Ψx, which commute with each other and thus are more applicable.

In this article, we focus on the gauge transformations of the constrained mKP (cmKP) hierarchy. The corresponding gauge transformations are required to keep not only the Lax equation but also the Lax operator [13,25,30]. For this, by selecting the special generating eigenfunction and adjoint eigenfunction, the elementary gauge transformation operators of the modified KP hierarchy TD(Φ) and TI(Ψ), become the ones in the constrained case. In fact, the Lax operator has m + 1 components after the gauge transformation, compared with m components at initial. Thus one of the components must be 0 to keep the original form of the Lax operator. Special generating functions of TD and TI must be chosen to satisfy this condition. Finally, the corresponding successive applications of TD and TI on the eigenfunction Φ and the adjoint eigenfunction Ψ are discussed.

This paper is organized in the following way. In Section 2, some basic facts about the mKP hierarchy are introduced. In Section 3, the gauge transformations of the constrained mKP hierarchy are studied. Then in Section 4, we discuss the successive applications of the gauge transformation operators TD and TI. At last in Section 5, some conclusions and discussions are given.

2. The mKP hierarchy

The mKP hierarchy in Kupershmidt-Kiso version [1820, 24] is defined as the following Lax equation

tnL=[(Ln/l)1,L],n=1,2,3,(2.1)
with the Lax operator L given by the pseudo-differential operator belowa
L=l+ul1l1+ul2l2+.(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 = ∑i aii, Ak = ∑ik aii and A<k = ∑i<k aii. 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 A f or A · f will denote the operator product of A and f, and * stands for the conjugate operation: (AB)* = B*A*, * = −, f* = f .

Similar to the case of the KP hierarchy [9,11], the Lax operator L for the mKP hierarchy can be expressed in terms of the dressing operator Z,

L=ZlZ1,(2.5)
where Z is given by
Z=z0+z11+z22+(z01exists).(2.6)

Then the Lax equation (2.1) is equivalent to

tnZ=(Ln/l)0Z=(ZnZ1)0Z.(2.7)

Define the wave function and the adjoint wave function [6, 29] 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 [29] below

resλw(t,λ)w*(t,λ)=1,
which is equivalent to the mKP hierarchy. Here resλi aiλ i = a−1.

It is proved in [29] that there exist two tau functions τ1 and τ0 for the mKP hierarchy in Kupershmidt-Kiso version such that

w(t,λ)=τ0(t[λ1])τ1(t)eξ(t,λ),(2.13)
w*(t,λ)=τ1(t+[λ1])τ0(t)λ1eξ(t,λ).(2.14)

By comparing (2.11) with (2.13), one can find

z0=τ0(t)τ1(t),z1z0=xlnτ0(t).(2.15)

The eigenfunction Φ and the adjoint eigenfunction Ψ of the mKP hierarchy [26] are defined in the identities below,

Φtn=(Ln/l)1(Φ),Ψtn=1(Ln/l)1*(Ψ).(2.16)

Note that the definition of the adjoint eigenfunction Ψ here is different from the one in [6]. For L to be the Lax operator (2.2) of the mKP hierarchy, let

L(1)=TLT1.(2.17)

If the following Lax equation

tnL(1)=[(L(1))1n/l,L(1)]
still holds, then T is called the gauge transformation operator of mKP hierarchy.

Lemma 2.1 ([6]).

If the pseudo-differential operator T satisfies

(TLn/lT1)1=T(Ln/l)1T1+TtnT1,(2.18)
then T is a gauge transformation operator of the mKP hierarchy.

It is proved in [6], there are two kinds of gauge transformation operators which can commute with each other. The specific forms are as follows:

  • Differential type

    TD(Φ)=(Φ1)x1Φ1,(2.19)

  • Integral type

    TI(Ψ)=Ψ11Ψx(2.20)
    where Φ is eigenfunction and Ψ is adjoint eigenfunction. It is important to note that the adjoint eigenfunction Ψ here is different from the one in [6], so the form of TI(Ψ) here is different.

Further, one can obtain the following lemma.

Lemma 2.2 ([6]).

Under the gauge transformation operator TD(Φ) and TI(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)=
TD(Φ) TD(Φ)Z∂−1 1/Φ )x/−1)x −1)x ·∫ ΦΨ1dx Φτ1 Φxτ12/τ0
TI(Ψ) TI(Ψ)Z∂ ∫ ΨxΦ1dx/Ψ 1x)x · Ψ Ψxτ02/τ1 τ0 · Ψ
Table I.

gauge transformations TD(Φ) and TI(Ψ)

The constrained mKP hierarchy [26] can be defined by imposing the following additional constraint on the Lax operator (2.2) of the mKP hierarchy,

L=l+i=1l1uii+i=1mΦi1Ψi.(2.21)

When l = 1, m = 1, (2.21) will become into

L=+ϕ1ψ.

Compared with the mKP’s Lax operator (see (2.2)), we can see that

u0=ϕψ,u1=ϕψx,u2=ϕψxx,

Then by (2.1) we know that

u0t2=ϕxxψ+2ϕψψxϕψxx+2ϕ2ψψx,ϕt2=ϕxx+2ϕψϕx,ψt2=ψxx+2ϕψψx;
and
u0t3=ϕxxxψ+ϕψxxx+3ψ2(ϕϕxx+(ϕx)2+ϕ3ψx)+3ϕ2(ψ3ϕx(ψx)2+(ϕx)2)3ϕϕxψψx,ϕt3=ϕxxx+3ϕψϕxx+3(ϕx)2ψ+3(ϕψ)2ϕx,ψt3=ψxxx3(ϕψ)xψx+3(ϕx)2+3(ϕψ)2ψx.

Remark

The relation between the KP hierarchy and the modified KP hierarchy can be extended to the constrained cases, which is shown in [26]. In fact, for the Lax operator of the constrained KP hierarchy

LKP=l+i=0l2vii+i=1mqi1ri,
and f be the eigenfunction of the constrained KP hierarchy corresponding to LKP, the transformed operator L˜=f1LKPf will be the Lax operator of the constrained mKP hierarchy, satisfying the following constraint
(L˜)<1=i=1m+1q˜i1r˜i
with
q˜i=f1qi,r˜i=fridx,i=1,2,,m
and
q˜m+1=f1L(f),r˜m+1=1.

3. Gauge transformation of the cmKP hierarchy

In this section, we will investigate the gauge transformations of the cmKP hierarchy. Different from the mKP hierarchy, the gauge transformation of constrained mKP must preserve the form of Lax operator. In order to discuss this problem more conveniently, here we need to introduce some basic lemmas on pseudo-differential operators.

Lemma 3.1 ([25]).

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

(Af1)<0=A0(f)1+A<0f1,(3.1)
(1gA)<0=1A0*(g)+1gA<0.(3.2)

Lemma 3.2.

For any pseudo-differential operator A, one has the following operator identities:

(TD(Φ)A1(TD(Φ))1)0=(TD(Φ)A1)(Φ)1Φ1,(3.3)
(TI(Ψ)A1(TI(Ψ))1)0=Ψ11(1(TI(Ψ)1)*(A1)*(Ψx)).(3.4)

Proof.

For (3.3), we can use the first expression in Lemma 3.1,

(TD(Φ)A1(TD(Φ))1)0=((Φ1)x1Φ1A11)<0((Φ1)x1Φ1A1Φ1Φ1)<0=((Φ1)x1Φ1A1Φ1)<0Φ1=(TD(Φ)A1)(Φ)1Φ1.

For (3.4), by the second formula in Lemma 3.1,

(T1(Ψ)A1(TI(Ψ))1)0=(Ψ11ΨxA11)<0+(Ψ11ΨxA1Ψx1Ψ)<0=(Ψ11ΨxA1dx)<0+(Ψ11ΨxA1Ψx1Ψ)<0=Ψ11(1(T1(Ψ)1)*(A1)*(Ψx)).

Lemma 3.3.

For any functions q and r, one has the following operator identities:

(TD(Φ)q1r(TD(Φ))1)0=TD(Φ)(q)1(1(TD(Φ)1)*)(r)(TD(Φ)q1r)(Φ)1Φ1,(3.5)
(T1(Ψ)q1r(T1(Ψ))1)0=T1(Ψ)(q)1(1(T1(Ψ)1)*)(r)Ψ11(1(T1(Ψ)1)*r1q)(Ψ).(3.6)

Proof.

(TD(Φ)q1r(TD(Φ))1)0=((Φ1)x1Φ1q1rΦ1(Φ1)x1)<0=((Φ1)x1Φ1q1rΦ(Φ111Φ1))<0=TD(Φ)(q)1rIII,(3.7)
where
III=((Φ1)x1Φ1q1(rrx)Φ1Φ1)<0=(TD(Φ)q1r)(Φ)1Φ1+TD(Φ)(q)1Φ1rxΦdx.

Substitute III into (3.7) and we can get (3.5).

As for (3.6),

(TI(Ψ)q1r(TI(Ψ))1)0=(Ψ11Ψxq1rΨx1Ψ1)<0=Ψ11qrΨIV+V,(3.8)
where
IV=(Ψ11Ψxq1rxΨx1Ψ)<0=TI(Ψ)(q)1rxΨx1ΨΨ11ΨxqdxrxΨx1Ψ.,
V=(Ψ11Ψxq1r)<0=TI(Ψ)(q)1rΨ11Ψxqdxr..

Due to

(1(T1(Ψ)1)*r1q)(Ψ)=qrΨ+ΨxqdxrxΨx1ΨΨxqdxr,
(3.6) can be derived after the substitution of IV and V into (3.8).

Proposition 3.1.

Under the gauge transformation TD(Φ), the initial Lax operator of the constrained mKP hierarchy

L(0)=L1(0)+i=1mΦi(0)1Ψi(0)
will become into
L(1)=TD(Φ)L(0)(TD(Φ))1=L1(1)+L0(1),
where
L0(1)=Φ0(1)1Ψ0(1)+i=1mΦi(1)1Ψi(1),Φ0(1)=(TD(Φ)L(0))(Φ),Ψ0(1)=Φ1,Φi(1)=TD(Φ)(Φi(0)),Ψi(1)=(1(TD(Φ)1)*)(Ψi(0)).

Proof.

By using (3.3)(3.5) and Lemma 2.2, the above results can be directly obtained.

Notice that there is an extra term in the integral part of the L0(1). To preserve the form of the integral part, we choose Φ that coincides with one of the original eigenfunctions that appear in the L0(0), e.g, Φ=Φ1(0). In this case one has Φ1(1)=0, and Φ0(1) and Ψ0(1) take over the roles of Φ1(1) and Ψ1(1). Here Φ0(1) and Ψ0(1) are still the (adjoint) eigenfunctions, which are proved in the next proposition.

Proposition 3.2.

Under the gauge transformation operator TD(Φ),

  1. (I)

    Φ0(1)=(TD(Φ)L(0))(Φ) is still the eigenfunction,

  2. (II)

    Ψ0(1)=Φ1 is the adjoint eigenfunction.

Proof.

  1. (I)

    Since

    tn(L(0)(Φ))=(tnL(0))(Φ)+L(0)tn(Φ)=[(L(0))1n/l,L(0)](Φ)+L(0)(L(0))1n/l(Φ)=(L(0))1n/lL(0)(Φ),

    From the above we can see L(0)(Φ) is the eigenfunction of L(0). Thus −(TD(Φ)L(0))(Φ) is an eigenfunction of L(1) according to Lemma 2.2.

  2. (II)

    For Ψ0(1)=Φ1 to be the adjoint eigenfunctions, we can prove by using (2.18).

    (1((L(1))n/l)1*)(Φ1)=1((Φ1)x1Φ1(L(0))1n/lΦ1(Φ1)x)*(Φ1)1((Φ1)x1Φ1)tnΦ1(Φ1)x)*(Φ1)x=1((Φ1)x1Φ(Φ1(Φ1)x1)tn)(Φ1)x=(Φ1)tn.

    Similarly, one can get the following proposition.

Proposition 3.3.

Under the gauge transformation TI(Ψ), the initial Lax operator of the constrained mKP hierarchy

L(0)=(L(0))1+i=1mΦi(0)1Ψi(0)
will become into
L(1)=T1(Ψ)L(0)(T1(Ψ))1=L1(1)+L0(1),
where
L0(1)=Φ0(1)1Ψ0(1)+i=1mΦi(1)1Ψi(1),Φ0(1)=Ψ1,Ψ0(1)=(1(TI(Ψ)1)*(L(0))*)(Ψ),Φi(1)=TI(Ψ)(Φi(0)),Ψi(1)=(1(TI(Ψ)*)1)(Φi(0)).

Proof.

It can be proved by (3.4), (3.6) and Lemma 2.2.

It is important to note here that the Lax operator has m + 1 components after transformation. In order to maintain the original form, one of the transformed components of L(1) must be 0. Therefore, by letting Ψ=Ψ1(0) in Proposition 8, one has Ψ1(1)=0. And Φ0(1) and Ψ0(1) take over the roles of Φ1(1) and Ψ1(1). Similarly, Φ1(0) and Ψ1(0) are still the (adjoint) eigenfunctions due to the proposition below.

Proposition 3.4.

Under the gauge transformation operator TI(Ψ),

  1. (I)

    Φ0(1)=Ψ1 is still the eigenfunction.

  2. (II)

    Ψ0(1)=(1(TI(Ψ)1)*(L(0))*)(Ψ) is adjoint eigenfunction.

Proof.

  1. (I)

    In order to prove Φ0(1)=Ψ1 is an eigenfunction, we just have to prove tnΨ1=(L(1))1n/l(Ψ1). In fact according to (2.18),

    (L(1))1n/l(Ψ1)=(Ψ11ΨxL(0)Ψx1Ψ)1n/l(Ψ1)=(Ψ11Ψx(L(0))1n/lΨx1Ψ)(Ψ1)+(Ψ11Ψx)tnΨx1Ψ(Ψ1)=(Ψ1)tn.

  2. (II)

    Due to

    tn((1(L(0))*)(Ψ))=(1(L(0))tn*)(Ψx)+(1(L(0))*)(Ψxtn)=(1((L(0))n/l)1*L(0)*)(Ψx)=(1((L(0))n/l)n/l*)((1(L(0))*)(Ψ)),

    Thus (−1(L(0))*)(Ψ) is the adjoint eigenfunction of L(0). Thus − (−1(TI(Ψ)−1)*(L(0))*)(Ψ) is adjoint eigenfunction of L(1) according to Lemma 2.2.

    Summarize the results above in the next proposition.

Proposition 3.5.

Under the gauge transformation operator TD(Φ1(0)) and TI(Ψ1(0)), the Lax operator of constrained mKP hierarchy

L(0)=L1(0)+i=1mΦi(0)1Ψi(0)
becomes into
L(1)=L1(1)+i=1mΦi(1)1Ψi(1).
  • Under the action of TD(Φ1(0)):

    Φ1(1)=(TD(Φ1(0))L(0))(Φ1(0)),Ψ1(1)=(Φ1(0))1,Φi(1)=TD(Φ1(k))(Φi(k)),Ψi(1)=1(TD(Φ1(0))1)*(Ψi(0)),i=2,,m.

  • Under the action of TI(Ψ1(0)):

    Φ1(1)=(Ψ1(0))1,Ψ1(1)=1(T1(Ψ1(0))1)*L(0)*(Ψ1(0))Φi(1)=TI(Ψ1(0))(Φi(0)),Ψi(1)=1(TI(Ψ1(0))*)1(Ψi(0)),i=2,,m.

Remark.

In fact, the special choices in above propostion of the eigenfunction and the adjoint eigenfunction, in order to preserve the forms of the constrained Lax operator, go back to the case of the constrained KP hierarchy in [1, 2].

By using Proposition 3.5, we can find the result below.

Proposition 3.6.

For the gauge transformation operator of the cmKP hierarchy, we have

TI(Ψ1(1))TD(Φ1(0))=TD(Φ1(1))TI(Ψ1(0))=1.

Proof.

We know Ψ1(1)=(Φ1(0))1 from Proposition 3.5, thus

TI(Ψ1(1))TD(Φ1(0))=(Ψ1(1))11(Ψ1(1))x((Φ1(0))1)x1(Φ1(0))1=Φ1(0)1((Φ1(0))1)x((Φ1(0))1)x1(Φ1(0))1=1.

Similarly, TD(Φ1(1))TI(Ψ1(0))=1..

4. The successive applications of TD and TI

From Proposition 3.6, we can know the pair of TD and TI will cancel in the products. Therefore, one can only consider the products of TD or TI. First, we recall the corresponding results of the mKP hierarchy [6]. Consider the following chain of the gauge transformation operators TD(q) and TI(r) of the mKP hierarchy.

LTD(q1)L(1)TD(q2(1))L(2)L(n1)TD(qn(n1))L(n)TI(r1(n))L(n+1)TI(r1(n+1))L(n+k1)TI(rk(n+k1))L(n+k).

Denote

T(n,k)(r1,r2,,rk;q1,q2,,qn)=TI(rk(n+k1))TI(r1(n))TD(qn(n1))TD(q2(1))TD(q1).(4.1)

The generalized Wronskian determinant [13] is needed in the next propositions, which is defined in the following form

IWk,nIWk,n(rk,rk1,,r1;q1,,qn)=|q1rkdxq2rkdxqnrkdxq1rk1dxq2rk1dxqnrk1dxq1r1dxq2r1dxqnr1dxq1q2qnq1xq2xqnxq1(nk1)q2(nk1)qn(nk1)|

In particular,

IW0,n=Wn(q1,q2,,qn),(4.2)
which is Wronskian determinant of q1, q2,...,qn. financial-disclosure

Lemma 4.1 ([6]).

T (n,0) and (T (n,0))−1 have the following forms

T(n,0)(q1,q2,,qn)=1Wn(q1,q2,,qn)|q1q2qn1q1xq2xqnxq(n)q2(n)qn(n)n|(4.3)
and
T(n,0)(q1,q2,,qn)1=(1)n1Wn(q1,q2,,qn)|q11q1q1(n2)q21q2q2(n2)qn1qnqn(n2)|.(4.4)

Lemma 4.2 ([6]).

T (0,k) and (T (0,k))−1 have the following forms

T(0,k)(r1,,rk)=(1)k1Wk(r1,,rk)|r1xr1xxr1(k1)1r1xr2xr2xxr2(k1)1r2xrkxrkxxrk(k1)1rkx|,(4.5)
and
T(0,k)(r1,,rk)1=|1r1xrkxr1xxrkxx()kr1(k+1)rk(k+1)|Wk(r1,,rk)Wk+1(1,r1,r2,,rk)2.(4.6)

Lemma 4.3 (Jacobi Expansion, [3, 13]).

For Wronskian determinants,

(Wn(f1,f2,,fn1,fn+1)Wn(f1,f2,,fn1,fn))x=Wn+1(f1,f2,,fn,fn+1)Wn1(f1,f2,,fn1)Wn(f1,f2,,fn1,fn)2.(4.7)

Using the gauge transformation operator TD, one can construct the following n-step gauge transformation :

L(0)TD(0)L(1)TD(1)L(2)TD(2)TD(n1)L(n),
where TD(i)=TD(Φ1(i)). According to the above formula, we can obtain the following proposition.

Proposition 4.1.

For the Lax operator of the constrained mKP hierarchy, under n steps of TD, we have

L(0)=(L(0))1+i=1mΦi(0)1Ψi(0),(4.8)
(i)Φ1(n)=(1)nWn+1(Φ1(0),η1,η2,,ηn)Wn(Φ1(0),η1,η2,,ηn1),(4.9)
(ii)Φi(n)=Wn+1(Φ1(0),η1,η2,,ηn,Φi(0))Wn(Φ1(0),η1,η2,,ηn1),(4.10)
(iii)Ψ1(n)=(1)n1Wn1(Φ1(0),η1,η2,,ηn2)Wn(Φ1(0),η1,η2,,ηn1),(4.11)
(iv)Ψi(n)=(1)n1IW1,n(Ψ1x(0);Φ1(0),η1,,ηn1)Wn(Φ1(0),η1,η2,,ηn1)dx,(4.12)
where ηk=(L(0))k(Φ1(0)), i = 2,3,...,m and k = 1,2,3,....

Proof.

  1. (i)

    According to proposition 3.5,

    Φ1(1)=TD(Φ1(0))(L(0)(Φ1(0)))=TD(Φ1(0))(η1)=η1(1),Φ1(2)=(1)2TD(Φ1(1))TD(Φ1(0))(L(0)2(Φ1(0)))=(1)2TD(Φ1(1))TD(Φ1(0))(η2)=(1)2η2(2),
    we can know Φ1(k)(1)kηk(k) by induction, thus TD(i)=TD(η1(i)).
    Φ1(n)=(1)nTD(n1)TD(n2)TD(1)TD(0)(L(0)n(Φ1(0)))=(1)nT(n,0)(Φ1(0),η1,,ηn1)(ηn).

    Therefore with the help of (4.3), one can obtain (4.9).

  2. (ii)

    Φi(n)=T(n,0)(Φ1(0),η1,,ηn1)(Φ1(0)).

    In the same way as (i), we can get (ii).

  3. (iii)

    By Proposition 3.5

    Ψ1(n)=(Φ1(n1))1=(1)n1Wn1(Φ1(0),η1,η2,,ηn2)Wn(Φ1(0),η1,η2,,ηn1).

  4. (iv)

    According to Proposition 3.5 and (4.4),

    Ψi(n)=1((T(n,0)1)*(Ψi(0)),
    we can get (iv).

    Similarly, we can construct another n-step gauge transformation using only TI :

    L(0)TI(0)L(1)TI(1)L(2)TI(2)TI(n1)L(n).
    where TI(i)=TD(Ψ1(i)). Under the successive application of TI, we have the following conclusions.

Proposition 4.2.

For the Lax operator of the constrained mKP hierarchy, under n steps of TI, we have

L(0)=(L(0))1+i=1mΦi(0)1Ψi(0),
(i)Ψ1(n)=Wn+1(Ψ1(0),η^1,η^2,,η^n)Wn+1(1,Ψ1(0),η^1,η^2,,η^n1),(ii)Ψi(n)=Wn+1(Ψ1(0),η^1,η^2,,η^n1,Ψi(0))Wn+1(1,Ψ1(0),η^1,η^2,,η^n1),(iii)Φ1(n)=Wn(1,Ψ1(0),η^1,η^2,,η^n2)Wn(Ψ1(0),η^1,η^2,,η^n1),(iv)Φi(n)=IW1,n(Φix(0);Ψ1x(0),η1x,,ηn1,x)Wn(Ψ1(0),η^1,,η^n1),
where i = 2, 3,...,m and
η^k=(L(0)*)k(Ψ1x(0))dx.k=1,2,3,

Proof.

According to Proposition 3.5,

Ψ1(1)=1(TI(Ψ1(0))1)*(1(L(0)*)(Ψ1(0)))=1(T1(Ψ1(0))1)*(η^1)=η^1(1),Ψ1(2)=(1)21(T1(Ψ1(1)TI(Ψ1(0))1)*(1(L(0)*)2(Ψ1(0)))=(1)21(TI(Ψ1(1)TI(Ψ1(0))1)*(η^2)=(1)2η^2(2),

One can know Ψ1(k)=(1)kη^k(k) by induction. Thus TI(i)=TI(η^i(i)). Then all the results can be proved with the same method as the one in Proposition 4.1, by using Lemma 4.2 and Lemma 4.3.

5. Conclusions and Discussions

The main results of this paper are as follows. First, we give two kinds of gauge transformations for the constrained modified KP hierarchy, which are summarized in Proposition 3.5. The corresponding gauge transformations should not only maintain the form of the Lax operator, but also keep the form of the Lax equation unchanged. For this, by selecting special generating eigenfunction and adjoint eigenfunction, the elementary gauge transformation operators of modified KP hierarchy TD(Φ)=(Φ1)x1Φ1 and TI(Ψ) = Ψ−1−1Ψx, become the ones in the constrained case, which are verified in Proposition 3.1 and Proposition 3.3 respectively. Second, we find that the two gauge transformations TD and TI can cancel each other. Therefore, in the Section 4, we can only discuss the successive applications of TD or TI. The specific forms of the successive applications of TD or TI are given in Proposition 4.1 and Proposition 4.2 respectively.

The selections of the special generating eigenfunction and adjoint eigenfunction in TD(Φ) and TI(Ψ) are very crucial, which can go back to Aratyn et al’s work in [1, 2] for the constrained KP case. Another kind of important choices is given by Oevel [25]. The eigenfunction φ, satisfying LKP(φ) = λ φ for some constant λ, is chosen as the generating eigenfunction for the constrained KP case in [25]. Willox et al’s work [30] provides one good rule (see (23) in [30]) of selecting eigenfunction in gauge transformation for the constrained KP case, containing the results of Aratyn and Oevel. Based upon these different choices, Willox et al [30] also considered an interesting binary transformation for the constrained KP hierarchy, which is the combination of the Darboux transformation of Aratyn’s type with the one of Oevel’s type.

By considering the relation between the constrained KP and mKP hierarchies [26], it will be very natural and interesting to ask what the relations between their gauge transformations are, and whether there are gauge transformations similar to those in [30] for the case of the constrained mKP hierarchy. For this, we will consider these questions in our future work.

This work is supported by China Postdoctoral Science Foundation (Grant No. 2016M591949) and Jiangsu Postdoctoral Science Foundation (Grant No. 1601213C). We thank anonymous referee for his/her useful comments and suggestions.

Footnotes

a

Usually, the parameter l should be 1. Here, we introduce the parameter l to express the constraint (see (23)) on the Lax operator in a convenient way. When l ≠ 1, the corresponding description of the modified KP hierarchy is equivalent to the usual case.

References

[9]E. Date, M. Kashiwara, and T. Miwa, Transformation groups for solition equations, M. Jimbo and T. Miwa (editors), Nonlinear integrable systems-classical theory and quantum theory, World Scientific, Singapore, 1983, pp. 39-119.
[12]J.S. He, Y.H. Li, and Y. Cheng, q-deformed KP hierarchy and q-deformed constrained KP hierarchy, SIGMA, Vol. 2, 2006, pp. 060.
[16]V. Kac and J. van de Leur. Equivalence of formulations of the MKP hierarchy and its polynomial tau-functions, arXiv: 1801. 02845
[28]T. Takebe, A note on the modified KP hierarchy and its (yet another) dispersionless limit, Lett. Math. Phys., Vol. 59, 2002, pp. 157-172.
Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
26 - 1
Pages
54 - 68
Publication Date
2021/01/06
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1080/14029251.2019.1544787How to use a DOI?
Copyright
© 2019 The Authors. Published by Atlantis 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  - Huizhan Chen
AU  - Lumin Geng
AU  - Na Li
AU  - Jipeng Cheng
PY  - 2021
DA  - 2021/01/06
TI  - Solving the constrained modified KP hierarchy by gauge transformations
JO  - Journal of Nonlinear Mathematical Physics
SP  - 54
EP  - 68
VL  - 26
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2019.1544787
DO  - 10.1080/14029251.2019.1544787
ID  - Chen2021
ER  -