Solutions of the constrained mKP hierarchy by Boson-Fermion correspondence

Huizhan Chen; Lumin Geng; Jipeng Cheng

doi:10.1080/14029251.2020.1700647

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Volume 27, Issue 2, January 2020, Pages 308 - 323

Solutions of the constrained mKP hierarchy by Boson-Fermion correspondence

Authors

Huizhan Chen, Lumin Geng, 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 3 June 2019, Accepted 6 October 2019, Available Online 27 January 2020.

DOI: 10.1080/14029251.2020.1700647 How to use a DOI?
Keywords: Hirota bilinear equation; Boson-Fermion correspondence; k-constrained mKP hierarchy; rational and soliton solutions
Abstract: In this paper, the Hirota bilinear equation of the constrained modified KP hierarchy is expressed as the vacuum expectation values of Clifford operators by using the free fermions method of mKP hierarchy. Then we mainly use the Boson-Fermion correspondence to solve the Hirota bilinear equation of the k-constrained mKP hierarchy. Further, by choosing special group elements in GL_∞, the corresponding rational and soliton solutions are given.
Copyright: © 2020 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

The modified KP hierarchy [14, 15] is introduced as a series of bilinear equations by using the Boson-Fermion correspondence, i.e.

resλ[τn(t−[λ−1];g)τn′(t+[λ−1];g)λn−n′eξ(t−t′,λ)]=0, n≥n′,(1.1)

where t = (t₁ = x, t₂, t₃,···) and [λ]=(λ,λ22,λ33,⋯). Here the tau functions are defined as follows

τn(t;g)=〈n|eH(t)g|n〉, g∈G,(1.2)

where

H(t)=∑l=1∞∑n∈𝕑φnφn+l*tl.

ϕ_n, φn*(n∈𝕑) are generators of the Clifford algebra A which satisfy

[φn,φm*]+=φnφm*+φm*φn=δm,n,[φn,φm]+=[φn*,φm*]+=0.(1.3)

G = {g ∈ 𝒜 ∃g⁻¹, gVg⁻¹ = V, gV^*g⁻¹ = V^*}, with V=∑n∈𝕑Cφn and V*=∑n∈𝕑Cφn*. 〈n| and |n〉 are the states of the charge n, which are defined as follows:

〈n|={〈0|φ−1⋯φn,n<0〈0|, n=0〈0|φ0*⋯φn−1*,n>0.,

|n〉={φn+1*⋯φ−1*|0〉,n<0|0〉, n=0φ−n+1⋯φ0|0〉,n>0..

And 〈0| and |0〉 are the vacuum states

φn|0〉=0,(n<0);φn*|0〉=0,(n≥0),〈0|φn=0,(n≥0);〈0|φn=0,(n<0),〈0|0〉=1.(1.4)

After the work above, people paid more attention to find the Lax equation forms. There are many versions of the mKP hierarchy [3, 7, 16–18, 21, 26]. All the forms are trying to generalize the Miura link between the mKdV and KdV equations to the KP cases. Here in this paper, we only consider the Kupershmidt-Kiso version [16–18,26]. In fact, the mKP hierarchy of Kupershmidt-Kiso version is corresponding to the bilinear equation [5]

resλ(λ−1eξ(t−t′,λ)τ0(t−[λ])τ1(t′+[λ−1]))=τ1(t)τ0(t′),(1.5)

which can be rewritten [22] from the original bilinear equation (1.1).

The wave function and the adjoint wave function [5, 14, 15] can be defined through the tau functions τ₁ and τ₀ in the following way

w(t,λ)=τ0(t−[λ−1])τ1(t)eξ(t,λ)=〈1|eH(t)φ(λ)g|0〉〈1|eH(t)g|1〉,(1.6)

w*(t,λ)=τ1(t+[λ−1])τ0(t)λ−1e−ξ(t,λ)=〈0|eH(t)φ*(λ)g|1〉λ〈0|eH(t)g|0〉,(1.7)

where ξ(t,λ)=∑j=1∞tjλj. Thus the bilinear equations (1.5) are equivalent to

resλw(t′,λ)w*(t,λ)=1.(1.8)

Then one can introduce two pseudo-differential operators Z and W such that the relations below hold

w(t,λ)=Z(eξ(t,λ)), w*(t,λ)=W(−e(t,λ)),(1.9)

with

Z0=z0+z1∂−1+z2∂−2+⋯, W=w1∂−1+w2∂−2+⋯.(1.10)

Here ∂ = ∂_x. The algebraic multiplication of ∂ⁱ with the multiplication operator f is given by the usual Leibnitz rule [8]

∂if=∑j≥0(ij)f(j)∂i−j, i∈𝕑,(1.11)

where f(j)=∂jf∂xj. For A=∑iai∂i, A≥k=∑i≤kai∂i, A<k=∑i<kai∂i and A_[k] = a_k. 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 or A · f will denote the operator product of A and f, and * stands for the conjugate operation: (AB)^* = B^*A^*, ∂^* = −∂, f^* = f.

Lemma 1.1

([12]). If we let A(x)=∑iai(x)∂xi and B(x′)=∑jbj(x′)∂x′j be two operators, then

A(x)B*(x)∂x(Δ0)=resλA(x)(exλ)⋅B(x′)(e−x′λ),(1.12)

where Δ⁰ = (x − x′)⁰ and

∂x−a(Δ0)={0,a<0,(x−x′)aa!,a≥0.(1.13)

From the bilinear equation (1.8), one can find W = (Z∂⁻¹)^* and the evolution equations of the operator Z as follows

∂tnZ=−(Z∂nZ−1)≤0Z.(1.14)

Here the operator Z is called the dressing operator or the wave operator. Then one can introduce the Lax operator of the mKP hierarchy [16–18, 26] in the way below

L=Z∂Z−1=∂+u0+u1∂−1+u2∂−2+u3∂−3+⋯,(1.15)

which satisfies the Lax equations

∂tnL=[(Ln)≥1,L], n=1,2,3,….(1.16)

The mKP hierarchy and its various extensions have many important integrable structures, such as tau function [5, 32], Hirota bilinear equation [2, 5], squared eigenfunction symmetries [5, 28, 29], additional symmetry [5, 33], Hamiltonian structures [3, 7, 17, 27], gauge transformation [1, 4, 6, 13, 23, 30, 34], and algebraic strucutre [21, 24] etc. In this paper, we mainly discuss the constrained mKP hierarchy, which can be viewed as the sub-hierarchy of the mKP hierarchy.

The k-constrained mKP hierarchy [2, 5, 29] is defined by imposing the following constraint on the Lax operator,

Lk=(Lk)≥1+∑i=1mqi∂−1ri∂,(1.17)

where q and r are the eigenfunction and the adjoint eigenfunction of the mKP hierarchy respectively, satisfying

∂tnqi=(Ln)≥1(qi), ∂tnri=−(∂(Ln)≥1∂−1)*(ri).(1.18)

Introduce two auxiliary functions ρ(t) and σ(t) such that

qi(t)=ρi(t)τ1(t), ri(t)=σi(t)τ0(t),(1.19)

Then we have the proposition below [2].

Proposition 1.1

([2]). The auxiliary functions σ(t), ρ(t), τ₁(t), and τ₀(t) satisfy the following bilinear equations:

∑i=1mρi(t)σi(t′)=resλ(λk−1τ0(t−[λ−1])τ1(t′+[λ−1])eξ(t−t′,λ)),(1.20)

ρi(t)τ0(t′)=resλ(λ−1τ0(t−[λ−1])ρi(t′+[λ−1])eξ(t−t′,λ)),(1.21)

σi(t′)τ1(t)=resλ(λ−1τ1(t′+[λ−1])σi(t−[λ−1])eξ(t−t′,λ)).(1.22)

Boson-Fermion correspondence [9–11, 14, 15, 19, 20, 25] describes the link between the free fermions and the Bosons, which is a very powerful tool to deal with the integrable hierarchy. By this method, the solutions for the constrained KP and BKP hierarchies are constructed from its bilinear representations [31,35]. In this paper, we will rewrite the Hirota bilinear equations of the constrained mKP hierarchy in the proposition above, in terms of the vacuum expectation values of Clifford operators by using Boson-Fermi correspondence [11, 25]. Then we mainly use the Boson-Fermion correspondence to solve the Hirota bilinear equation of the k-constrained mKP hierarchy. Further, by choosing special group elements in GL_∞, the corresponding rational and soliton solutions are given.

This paper is organized in the following way. Rational solutions for the vector k-constrained mKP hierarchy are derived in Section 2. Section 3 is devoted to the soliton solutions of k-constrained mKP hierarchy. At last, some conclusions and discussions are given in Section 4.

2. Rational solutions for the vector k-constrained mKP hierarchy

Lemma 2.1

(Wick’s theorem [25]). For ϕ₁,···ϕ_r ∈ V ⊕ V^*,

〈φ1⋯φr〉={0,if r is odd,∑σsign(σ)〈φσ(1)φσ(2)〉⋯〈φσ(r−1)φσ(r)〉,if r is even.(2.1)

where 〈·〉 = 〈vac|·|vac〉 and sign(σ) is the sign of a permutation; the sum runs over all permutations σ satisfying σ(1) < σ(2),···, σ(r − 1) < σ(r) and σ(1) < σ(3) < ··· < σ(r − 1), in other words, over all ways of grouping the ϕ_i into pairs.

Denote φ(λ)=∑n∈𝕑φnλn and φ*(λ)=∑n∈𝕑φn*λ−n. Define the t-evolution of an operator a as a(t) = e^H(t)ae^−H(t). Then

Lemma 2.2 ([11]).

The following relations hold

H(t)|vac〉=0,(2.2)

eH(t)φ(λ)e−H(t)=eξ(t,λ)φ(λ),(2.3)

eH(t)φ*(λ)e−H(t)=e−ξ(t,λ)φ*(λ),(2.4)

〈vac|φn*(t)φm(t)|vac〉=∑i≥0pi−n(−t)pm−i(t),(2.5)

where p_n(t) is the Schur polynomial, determined by eξ(t,λ)=∑n=0∞pn(t)λn.

From this lemma, one can obtain

eH(t)φne−H(t)=∑i=0∞φn−ipi(t),(2.6)

eH(t)φn*e−H(t)=∑i=0∞φn+i*pi(−t).(2.7)

The following differential operators of infinite order are called vertex operators.

X(λ)=eξ(t,λ)e−ξ(∂˜,λ−1)=∑i∈𝕑Xi(t,∂∂t)λi,(2.8)

X*(λ)=e−ξ(t,λ)eξ(∂˜,λ−1)=∑i∈𝕑Xi*(t,∂∂t)λ−i,(2.9)

where X_i, Xi* are given by

Xi=∑n≥0pn+i(t)pn(−∂˜), Xi*=∑n≥0pn−i(−t)pn(∂˜)(2.10)

and ∂˜=(∂1,12∂2,13∂3,…).

Lemma 2.3 ([11]).

The vertex operators (2.8) and (2.9) have the following properties:

〈n|eH(t)φ(λ)=λn−1X(λ)〈n−1|eH(t),(2.11)

〈n|eH(t)φ*(λ)=λ−nX*(λ)〈n+1|eH(t).(2.12)

From Ref. [11], for any g ∈ G, it satisfies the following commutation relations with free fermion operators,

φng=∑i∈𝕑gφi(a−1)i,n, φn*g=∑i∈𝕑gφi*ai,n.(2.13)

Proposition 2.1.

Let Γk=∑n∈𝕑φnφn+k*, if g ∈ G satisfies the condition

g−1Γkg=∑i,j∈𝕑fi,jφiφj*,(2.14)

with fi,j=∑l=1mdi(l)ej(l) for i ≥ 0, j ≤ 0, then

τ0(t)=〈0|eH(t)g|0〉,(2.15)

τ1(t)=〈1|eH(t)g|1〉,(2.16)

ρi(t)=〈1|eH(t)g∑n≥0dn(i)φn|0〉,(2.17)

σi(t)=〈0|eH(t)g∑n≤0en(i)φn*|1〉.i=1,2,…,m(2.18)

satisfy the bilinear equations in Proposition 1.1.

Proof.

We first prove that Eq. (2.14), (2.16), (2.17) and (2.18) satisfy (1.20). According to Lemma 2.3

Due to g ∈ G, we have

φig=∑m∈𝕑gφm(a−1)m,i, φi*g=∑m∈𝕑gφm*ai,m,(2.20)

further,

g−1Γkg=g−1∑l∈𝕑φlφl+k*g=∑l∈𝕑g−1φlgg−1φl+k*g=∑l,i,j∈𝕑g−1gφi(a−1)i,lg−1gφj*al+k,j=∑l,i,j∈𝕑φiφj*(a−1)i,lal+k,j.(2.21)

According to the assumption of the theorem

∑n∈𝕑(a−1)i,nan+k,j=fi,j=∑l=1mdi(l)ej(l), i≥0,j≤0,(2.22)

by using (1.4) and (2.20), (2.19) can be further written as

So (1.20) is verified. And then we will prove (1.22). By using Lemma 2.3

(2.24) can be rewritten as

where we haved used (1.3), (1.4) and (2.20). Therefore (1.22) is proven. In the same way, it is easy to prove (1.21).

Remark 2.1.

Note that these results are different from the ones in the KP case [35]. The major differences are as follows. 1) The subscript j in ej(l) can be zero in the mKP case, while zero is forbidden in the KP case. 2) The expressions of σ_i are different. Since zero is allowed in subscript j of ej(l), f₀₀ will appear and it will bring much difficulty when discussing the rational solutions. So in the lemma below, in order to avoid these problems, we separate the zero parts from the nonzero parts in the expressions of the group elements in GL_∞.

In order to find rational solutions of the k-constrained mKP hierarchy more conveniently by using the proposition 2.1, the following lemma is needed.

Lemma 2.4

Let g=eaφ0*φ0+∑n=1Nbnφin*φjn=eY, where a and b_n are constant, i_n < 0, j_n > 0 for n = 1, 2,...,N, then

g−1Γkg=Γk+∑m,n=1Nbmbnδin+k,jnφim*φjn+∑n=1Nbn(φin*φjn−k−φim+k*φjn)+(1−ea)(φk*φ0−∑n=1Nbnδjn,kφin*φ0)+(1−e−a)(φ0*φ−k+∑n=1Nbnδ−in,kφ0*φjn).(2.26)

Proof.

Direct calculation shows

(adY)Γk=a(φk*φ0−φ0*φ−k)+∑n=1Nbn(φin+k*φjn−φin*φjn−k),(adY)2Γk=a2(−φk*φ0−φ0*φ−k)+2a∑n=1Nbn(δjn,kφin*φ0+δk,−inφ0*φjn)+2∑m,n=1Nbmbnδin+k,jmφim*φjn,(adY)3Γk=a3(φk*φ0−φ0*φ−k)+3a2∑n=1Nbn(−δjn,kφin*φ0+δk,−inφ0*φjn),⋯(2.27)

Further calculation by means of mathematical induction method,

(adY)mΓk=am((−1)m−1φk*φ0−φ0*φ−k)+mam−1∑n=1Nbn((−1)mδjn,kφin*φ0+δk,−inφ0*φjn), m≥3.(2.28)

Then by Baker-Campbell-Hausdorff-formula

g−1Γkg=e−YΓkeY=∑m=0∞(−1)m1m!(adY)mΓk=Γk+∑m=1∞(−1)mam((−1)m−1φk*φ0−φ0*φ−k)m!+∑m=2∞(−1)mmam−1∑n=1Nbn((−1)mδjn,kφin*φ0+δk,−inφ0*φjn*)m!+∑m,n=1Nbmbnδin+k,jmφim*φjn−∑n=1Nbn(φin+k*φjn−φin*φjn−k).(2.29)

Using the Taylor’s formula to calculate the above formula, one can get (2.26).

To better understand Proposition 2.1, let’s use Lemma 2.4 to give an example.

Example 2.1.

We consider the case where m = 1 in (1.20) and N = 1 in Lemma 2.4, then

g−1Γkg=Γk+b12δi1+k,j1φi1*φj1+(ea−1)b1δj1,kφi1*φ0+(1−e−a)b1δ−i1,kφ0*φj1+b1φi1*φj1−k−b1φi1+k*φj1+(1−ea)φk*φ0+(1−e−a)φ0*φ−k.(2.30)

By direct calculation using the mathematical induction hypothesis one can obtain

eY=eaφ0*φ0+b1φi1*φj1=1+(ea−1)φ0*φ0+b1φi1*φj1+(ea−1)b1φ0*φ0φi1*φj1.(2.31)

If i₁ < 0 and j₁ > 0, there are five cases to be discussed.

(1)
When i₁ + j₁ = k, by using Proposition 2.1 we find fj1,i1=−b12. So we may choose d_j₁ = −1, ei1=b12. Then by Lemma 2.1–Lemma 2.3 and Eqs.(2.8)–(2.10), we obtain the solutions of (1.20)–(1.22) are
τ0(t)=〈0|eH(t)eaφ0*φ0+b1φi1*φj1|0〉=〈0|eH(t)(1+(ea−1)φ0*φ0+b1φi1*φj1+(ea−1)b1φ0*φ0φi1*φj1)|0〉=1+(ea−1)〈φ0*(t)φ0(t)〉+b1φi1*(t)φj1(t)+(ea−1)b1(〈φ0*(t)φ0(t)〉〈φi1*(t)φj1(t)〉−〈φ0*(t)φj1(t)〉〈φi1*(t)φ0(t)〉)=ea+eab1∑n≥0pn−i1(−t)pj1−n(t)−(ea−1)b1∑n≥0pn(−t)pj1−n(t)p−i1(−t),(2.32)

τ1(t)=〈1|eH(t)eaφ0*φ0+b1φi1*φj1|1〉=〈1|eH(t)|1〉+(ea−1)〈1|eH(t)φ0*φ0|1〉+b1〈1|eH(t)φi1*φj1|1〉=(ea−1)b1〈1|eH(t)φ0*φ0φi1*φj1|1〉=X0〈0|eH(t)|0〉+(ea−1)〈1|eH(t)φ0*φ0φ0|0〉−b1Xj1〈0|eH(t)φi1*|1〉+(ea−1)b1〈1|eH(t)φ0*φ0φi1*φj1φ0|0〉=1−b1Xj1p−i1(−t),(2.33)

ρ1(t)=−〈1|eH(t)eaφ0*φ0+b1φi1*φj1φj1|0〉=−〈1|eH(t)φj1|0〉−(ea−1)〈1|eH(t)φ0*φ0φj1|0〉=−Xj1〈0eH(t)|0〉−(ea−1)〈1|eH(t)(1−φ0φ0*)φj1|0〉=−pj1(t)−(ea−1)Xj1〈0|eH(t)|0〉+(ea−1)Xj1〈0|eH(t)φ0φ0*|0〉=−eapj1(t),(2.34)

σ1(t)=〈0|eH(t)eaφ0*φ0+b1φi1*φj1b12φi1*|1〉=b12〈0|eH(t)φi1*|1〉+b12(ea−1)〈0|eH(t)φ0*φi1*|1〉+b13〈0|eH(t)φi1*φj1φi1*|1〉+b13(ea−1)〈0|eH(t)φ0*φ0φi1*φj1φi1*|1〉=b12〈0|eH(t)φi1*|1〉=b12p−i1(−t).(2.35)
(2)
When j₁ = k and −i₁ ≠ k, we find f_0,i₁ = b₁(1 − e^a). So one has d₀ = b₁, e_i₁ = −e^a. Then the solutions of (1.20)–(1.22) are
τ0(t)=ea+eab1∑n≥0pn−i1(−t)pj1−n(t)−(ea−1)b1∑n≥0pn(−t)pj1−n(t)p−i1(−t),(2.36)

τ1(t)=1−b1Xj1p−i1(−t),(2.37)

ρ1(t)=〈1|eH(t)eaφ0*φ0+b1φi1*φj1b1φ0|0〉=b1〈1|eH(t)φ0|0〉+b12〈1|eH(t)φi1*φj1φ0|0〉=b1X0〈0|eH(t)|0〉−b12Xj1〈0|eH(t)φi1*φ0|0〉=b1−b12Xj1pi1(−t),(2.38)

σ1(t)=〈0|eH(t)eaφ0*φ0+b1φi1*φj1(1−ea)φi1*|1〉=(1−ea)p−i1(−t).(2.39)
(3)
When j₁ ≠ k and −i₁ = k, one has d_j₁ = b₁ and e₀ = e^−a − 1. The solutions of (1.20)–(1.22) are
τ0(t)=ea+eab1∑n≥0pn−i1(−t)pj1−n(t)−(ea−1)b1∑n≥0pn(−t)pj1−n(t)p−i1(−t),(2.40)

τ1(t)=1−b1Xj1p−i1(−t),(2.41)

ρ1(t)=〈1|eH(t)eaφ0*φ0+b1φi1*φj1b1φj1|0〉=b1eapj1(t),(2.42)

σ1(t)=〈0|eH(t)eaφ0*φ0+b1φi1*φj1(e−a−1)φ0*|1〉=(1−ea)〈φ0*(t)φ0(t)〉+(1−ea)b1(〈φ0*(t)φ0(t)〉〈φi1*(t)φj1(t)〉−〈φ0*(t)φj1(t)〉〈φi1*(t)φ0(t)〉)=1−ea+(1−ea)b1(−p−i1(−t)∑n≥0pn(−t)pj1−n(t)+∑m≥0pm−i1(−t)pj1−m(t)).(2.43)
(4)
When j₁ = k and −i₁ = k, one has f_0,i₁ = b₁(1 − e^a), f_j₁,0 = b₁(1 − e^−a). So we may choose d₀ = 1 − e^a, d_j₁ = b₁, e_i₁ = b₁, e₀ = 1 − e^−a. Then
τ0(t)=ea+eab1∑n≥0pn−i1(−t)pj1−n(t)−(ea−1)b1∑n≥0pn(−t)pj1−n(t)p−i1(−t),(2.44)

τ1(t)=1−b1Xj1p−i1(−t),(2.45)

ρ1(t)=〈1|eH(t)eaφ0*φ0+b1φi1*φj1((1−ea)φ0+b1φj1)|0〉=(1−ea)(1−b1Xj1p−i1(−t))+b1eapj1(t),(2.46)

σ1(t)=〈0|eH(t)eaφ0*φ0+b1φi1*φj1((1−e−a)φ0*+b1φi1*)|0〉=(1−ea)1+b1p−i1(−t)+(1−ea)b1(∑n≥0pn−i1(−t)pj1−n(t)−p−i1(−t)∑m≥0pm(−t)pj1−m(t)).(2.47)
(5)
When j₁ < k, −i₁ < k, j₁ ≠ −i₁ + k, or j₁ > k, −i₁ < k, or j₁ ≠ k, i₁ < −k, there are no corresponding solutions.

3. Soliton solutions of k-constrained mKP hierarchy

In this part, we mainly use the Boson-Fermion correspondence to find the soliton solution of mKP hierarchy.

Proposition 3.1 (Boson-Fermion correspondence, [11]).

There exists an isomorphism Φ from the Fermionic Fock 𝒡 = 𝒜 |0〉 to the Bosonic Fock space 𝒝 = ℂ[t₁, t₂,...,u, u⁻¹], which is defined in the way below,

Φ(a|vac〉)=∑l∈𝕑〈l|eH(t)a|vac〉ul,(3.1)

where a|vac〉 ∈ 𝒡. Then the actions of ϕ(λ) and ϕ* (λ) on 𝒡 can be realised on 𝒝 as follows,

Φ(φ(λ)a|0〉)=X(λ)S(λ)Φ(a|0〉), Φ(φ*(λ)a|0〉)=X*(λ)S*(λ)Φ(a|0〉).(3.2)

Here

(S(k)f)(t,u)=uf(t,ku), (S*(k)f)(t,u)=ku−1f(t,k−1u).(3.3)

Remark 3.1.

Denote A(m) as the subspace of 𝒜 with the charge m operators. If a ∈ A(m), then only the term with l = m survives in the expression of Φ(a|vac〉).

Let

X(p,q)=eξ(x,p)−ξ(x,q)e−ξ(∂˜,p−1)+ξ(∂˜,q−1),(3.4)

Then we have

X(p)X*(q)=11−q/pX(p,q).(3.5)

Corollary 3.1 ([11]).

For a ∈ A(0),

Φ(φ(p)φ*(q)a|0〉)=qp−qX(p,q)〈0|eH(t)a|0〉.(3.8)

For a ∈ A(1),

Φ(φ(p)φ*(q)a|0〉)=upp−qX(p,q)〈1|eH(t)a|0〉.(3.11)

After the preparation above, one can obtain the following conclusion.

Lemma 3.1 ([35]).

Let g=e∑i,j=1Nbijφ(pi)φ*(qj), with b_ij has the form ∏l,s=1N(pl−qs)cij, p_i, q_j, c_ij are given constant and p_i ≠ q_j. Then

g−1Γkg=Γk+∑l,m∈𝕑∑i,j=1Nbijpilqj−m(pik−qjk)φlφm*.(3.12)

Further if bij=∑l=1mdi(l)ej(l)pik−qjk, i, j = 1, 2···N, by Lemma 3.1 we find easily g−1Γkg=Γk+∑l=1m∑i=1Ndi(l)φ(pi)∑j=1Nej(1)φ*(qj).

Proposition 3.2.

The soliton solutions of (1.20)–(1.22) in Proposition 1.1 are listed as follows:

τ0(t)=〈0|eH(t)g|0〉,(3.13)

τ1(t)=〈1|eH(t)g|1〉,(3.14)

ρi(t)=〈1|eH(t)g∑n=1Ndn(i)φ(pn)|0〉,(3.15)

σi(t)=〈0|eH(t)g∑n=1Nen(i)φ*(qn)|1〉,i=1,2,…,m.(3.16)

Proof.

It is easy to prove by using the conclusion of proposition 2.1.

Proposition 3.3.

The Eqs. (3.13)–(3.16) formulas in the proposition 3.2 can be expressed as the following forms respectively:

τ0(t)=1+∑n=1N∑1≤i1<i2<⋯<in≤N,1≤ju≤N,ju≠jv,u≠vb˜i1j1b˜i2j2⋯b˜injn×∏s<l(pis−pil)(qjs−qjl)(pis−qil)(qjs−pjl)e∑α=1n(ξ(t,piα)−(ξ(t,qjα),(3.17)

τ1(t)=1+∑n=1N∑1≤i1<i2<⋯<in≤N,1≤ju≤N,ju≠jv,u≠vb˜i1j1b˜i2j2⋯b˜injnpi1⋯pinqj1⋯qjn×∏s<l(pis−pil)(qjs−qjl)(pis−qil)(qjs−pjl)e∑α=1n(ξ(t,piα)−ξ(t,qjα)),(3.18)

ρi(t)=∑n=1Ndn(i)eξ(t,pn)+∑j=1N∑n=1N−1∑1≤i1<i2<⋯<in≤N1≤ju≤N,ju≠jv,u≠vju≠jdj(i)b˜i1j1b˜i2j2⋯b˜injn×Πs<l(pis−pil)(qjs−qjl)(pis−qil)(qjs−pjl)Πα=1n(pj−piα)(pj−qjα)e∑β=1n(ξ(t,piβ)+ξ(t,pj)−ξ(t,qjβ)),(3.19)

σi(t)=∑n=1Nen(i)eξ(t,qn)+∑j=1N∑n=1N−1∑1≤i1<i2<⋯<in≤N,1≤ju≤N,ju≠jv,u≠vju≠jej(i)b˜i1j1b˜i2j2⋯b˜injnpi1⋯pinqj1⋯qjn×Πs<l(pis−pil)(qjs−qjl)(pis−qil)(qjs−pil)Πα=1n(pj−qjα)(piα−qj)e∑β=1n(ξ(t,piβ)−ξ(t,qj)−ξ(t,qjβ)),(3.20)

where b˜ij=bijqjpi−qj, i, j = 1,2,...,N.

Proof.

We first prove (3.18). Using Lemma 3.1, one can find

τ1(t)=〈1|eH(t)g|1〉=u−1Φ(gφ0|0〉)=u−1Φ(Πi,j=1N(1+b˜ijpi−qjqjφ(pi)φ*(qj)⋅φ0|0〉)).(3.21)

Then according to (3.11) of Corollary 3.1, the above equation can be written as

u−1Πi,j=1N(1+b˜ijpiqjX(pi,qj))〈1|eH(t)φ0|0〉u=Πi,j=1N(1+b˜ijpiqjX(pi,qj))X0〈0|eH(t)φ0|0〉=Πi,j=1N(1+b˜ijpiqjX(pi,qj))⋅1=1+∑n=1N∑1≤i1<i2<⋯<in≤N,1≤ju≤N,ju≠jv,u≠vb˜i1j1b˜i2j2⋯b˜injnpi1⋯pinqj1⋯qin×Πs<l(pis−pil)(qjs−qjl)(pis−qjl)(qjs−pil)e∑α=1n(ξ(t,piα)−ξ(t,qjα)).(3.22)

Similarly, by lemma 3.1 and Corollary 3.1

σm(t)=Φ(g∑j=1Nej(m)φ*(qj)|1〉)=∏i,j=1N(1+b˜ijX(piqj))Φ(∑l=1Nel(m)φ*(ql)φ0|0〉)=∏i,j=1N(1+b˜ijX(piqj))∑l=1Nel(m)X*(ql)S*(ql)Φ(φ0|0〉=∏i,j=1N(1+b˜ijX(piqj))∑l=1Nel(m)X*(ql)S*(ql)⋅X0〈0|eH(t)|0〉u=∏i,j=1N(1+b˜ijX(piqj))∑l=1Nel(m)e−ξ(t,ql)⋅1=∑n=1Nen(m)e−ξ(t,qn)+∑j=1N∑n=1N−1∑1≤i1<i2<⋯in≤N,1≤ju≤N,ju≠jv,u≠vju≠jej(m)b˜i1j1b˜i2j2⋯b˜injnpj1⋯pjnqj1⋯qjn×Πs<l(pis−pil)(qjs−qjl)(pis−qjl)(qjs−pil)Πα=1n(pj−qjα)(piα−qj)e∑β=1n(ξ(t,piβ)−ξ(t,qj)−ξ(t,qiβ)).(3.23)

The proof method of (3.17) and (3.19) is the same as that of (3.18), and no more description is given here.

Remark 3.2.

If N = m in Proposition 3.2 we may choose di(l)=bil(pik−qlk) and el(j)=δlj. Accordingly, the solutions of the Hirota bilinear equation can be simplified to

τ0(t)=〈0|eH(t)g|0〉,(3.24)

τ1(t)=〈1|eH(t)g|1〉,(3.25)

ρi(t)=∑j=1mbij(pik−qjk)〈1|eH(t)gφ(pi)|0〉,(3.26)

σi(t)=〈0|eH(t)gφ*(qi)|1〉,i=1,2,…,m.(3.27)

Example 3.1.

For m = N = 2, b₁₁ ≠ 0, b₂₂ ≠ 0, b₁₂ = b₂₁ = 0, we have two-soliton solutions of (1.20)–(1.22):

τ0(t)=1+b11q1p1−q1eξ(t,p1)−ξ(t,q1)+b22q2p2−q2eξ(t,p2)−ξ(t,q2)+b11b22q1q2(p1−p2)(q1−q2)(p1−q1)(p1−q2)(q1−p2)(p2−q2)eξ(t,p1)+ξ(t,p2)−ξ(t,q1)−ξ(t,q2),(3.28)

τ1(t)=1+b11q1p1−q1eξ(t,p1)−ξ(t,q1)+b22p2p2−q2eξ(t,p2)−ξ(t,q2)+b11b22p1p2(p1−p2)(q1−q2)(p1−q1)(p1−q2)(q1−p2)(p2−q2)eξ(t,p1)+ξ(t,p2)−ξ(t,q1)−ξ(t,q2),(3.29)

ρ1(t)=b11(p1k−q1k)(eξ(t,p1)+b22q2(p1−p2)(p1−q2)(p2−q2)eξ(t,p1)+ξ(t,p2)−ξ(t,q2)),(3.30)

ρ2(t)=b22(p2k−q2k)(eξ(t,p2)+b11q1(p1−p2)(p1−q1)(p2−q1)eξ(t,p1)+ξ(t,p2)−ξ(t,q1)),(3.31)

σ1(t)=e−ξ(t,q1)−b22p2(q2−q1)(p2−q1)(p2−q2)eξ(t,p2)−ξ(t,q1)−ξ(t,q2),(3.32)

σ2(t)=e−ξ(t,q2)−b11p1(q2−q1)(p1−q1)(p1−q2)eξ(t,p1)−ξ(t,q1)−ξ(t,q2).(3.33)

4. Conclusions and Discussions

In this paper, we construct the solutions of the constrained mKP hierarchy through the bilinear representation and the free fermion operators. The corresponding solutions are expressed in terms of the vacuum expectation value of the Clifford operators, which are presented in Proposition 2.1. Then by choosing g=eaφ0*φ0+∑n=1Nbnφin*φjn=eY with i_n < 0 and j_n > 0, some examples of rational solutions are given. At last, by selecting g=e∑i,j=1Nbijφ(pi)φ*(qj), the corresponding soliton solutions are obtained, which are summarized in Proposition 3.2 and Proposition 3.3.

Just as we know, the KdV hierarchy is corresponding to the affine Lie algebra sl^2 [14, 25]. But the algebraic structures of the constrained mKP hierarchy are unknown. The results in this paper are expected to be helpful for the understanding the algebraic structures of the constrained mKP hierarchy.

Acknowledgments

This work is supported by China Postdoctoral Science Foundation (Grant No. 2016M591949) and Jiangsu Postdoctoral Science Foundation (Grant No. 1601213C).

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 27 - 2
Pages: 308 - 323
Publication Date: 2020/01/27
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TY  - JOUR
AU  - Huizhan Chen
AU  - Lumin Geng
AU  - Jipeng Cheng
PY  - 2020
DA  - 2020/01/27
TI  - Solutions of the constrained mKP hierarchy by Boson-Fermion correspondence
JO  - Journal of Nonlinear Mathematical Physics
SP  - 308
EP  - 323
VL  - 27
IS  - 2
SN  - 1776-0852
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