Bilinear identities for the constrained modified KP hierarchy

Huizhan Chen; Lumin Geng; Na Li; Jipeng Cheng

doi:10.1080/14029251.2019.1591721

<Previous Article In Issue

Download article (PDF)

Next Article In Issue>

Volume 26, Issue 2, March 2019, Pages 240 - 254

Bilinear identities for the constrained modified KP hierarchy

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 19 September 2018, Accepted 29 November 2018, Available Online 6 January 2021.

DOI: 10.1080/14029251.2019.1591721 How to use a DOI?
Keywords: the constrained mKP hierarchy; the bilinear identities; tau functions
Abstract: In this paper, we mainly investigate an equivalent form of the constrained modified KP hierarchy: the bilinear identities. By introducing two auxiliary functions ρ and σ, the corresponding identities are written into the Hirota forms. Also, we give the explicit solution forms of ρ and σ.
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

Recently, people have paid much attention to the modified Kadomtsev-Petviashvili (mKP) hierarchy [10, 12] in the field of integrable systems, such as the gauge transformations [2, 21], additional symmtries [3], the squared eigenfunction symmetries [3, 19, 20] and the tau functions [3, 25]. There are many versions [4, 6, 11, 13–15, 21] of the mKP hierarchy, which are all trying to generalize the Miura link [22] between KdV and mKdV to the KP case. In this paper, we will only consider the Kupershmidt-Kiso version [13– 15, 21]. By considering different reduction conditions on the Lax operator L, one can obtain the different kinds of sub-hierarchies. One of the most important sub-hierarchies is called the constained mKP (cmKP) hierarchy (see (2.16) in Section 2) [3, 20], which is a generalization of the reduction procedure from the mKP hierarchy to the generalized mKdV hierarchy. In this paper, we will discuss the bilinear formulations of the cmKP hierarchy.

The bilinear identity [7,10] is an important equivalent formulation of the integrable hierarchies. From the bilinear identity, one can easily obtain the Hirota equations [7, 9, 10]. And also it is very helpful in the discussion of the tau functions [7, 10]. There is much research on the bilinear identity of different integrable hierarchies, for example, the constrained KP and BKP hierarchies [5, 17, 23, 24], the extended bigraded Toda hierarchy [16, 18], the integrable hierarchies with with self-consistent sources [8,26]. In this paper, we firstly derive three bilinear identities, from the evolution equations of the (adjoint) wave functions and the constraint on the Lax operator. Then we show that these three bilinear identities can fully characterize the cmKP hierarchy. By introducing two auxiliary functions ρ and σ, the bilinear identities are written into the Hirota forms. At last, we give some solutions for ρ and σ.

This paper is organized in the following way. In Section 2, some basic facts about the mKP hierarchy are introduced. The bilinear identities for the cmKP hierarchy are derived in Section 3. Section 4 is devoted to the Hirota’s bilinear equations of the corresponding tau-functions and the explicit forms of ρ and σ. At last, some conclusions and discussions are given in Section 5.

2. Reviews of the mKP hierarchy

The definition of the mKP hierarchy is based on the theory of pseudo-differential operators, so we firstly introduce the knowledge of pseudo-differential operators [7, 14, 21]. The algebraic multiplication of ∂ⁱ with the multiplication operator f is given by the usual Leibnitz rule

∂if=∑j≥0(ij)f(j)∂i−j, i∈𝕑,(2.1)

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.

The mKP hierarchy in Kupershmidt-Kiso version [2, 13–15, 21] is defined as the following Lax equation

Ltn=[(Ln)≥1,L], n=1,2,3,…(2.2)

with the Lax operator L given below

L=∂+u0+u1∂−1+u2∂−2+u3∂−3+⋯.(2.3)

Here ∂ = ∂_x and u_i = u_i(t₁ = x, t₂,...). The Lax operator L of the mKP hierarchy can be expressed in terms of the dressing operator Z,

L=Z∂Z−1,(2.4)

where Z is given by

Z=z0+z1∂−1+z2∂−2+⋯(z0−1 exists).(2.5)

Then the Lax equation (2.2) is equivalent to

Ztn=−(Ln)≤0Z=−(Z∂nZ−1)≤0Z.(2.6)

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

w(t,λ)=Z(eξ(t,λ))=(z0+z1λ−1+z2λ−2+⋯)eξ(t,λ),(2.7)

w*(t,λ)=(Z−1∂−1)*(e−ξ(t,λ))=(z0−1+z1*λ−1+z2*λ−2+⋯)λ−1e−ξ(t,λ),(2.8)

where

ξ(t,λ)=xλ+t2λ2+t3λ3+⋯.(2.9)

Then w(t, λ) and w^*(t, λ) satisfy the bilinear identity [3, 25] below

resλw(t′,λ)w*(t,λ)=1,(2.10)

which is equivalent to the mKP hierarchy. Here resλ∑iaiλi=a−1.

The Lax equation (2.2) is compatible with the linear system

Lw(t,λ)=λw(t,λ), ∂tnw(t,λ)=(Ln)≥1(w(t,λ))(2.11)

and

(∂L∂−1)*w*(t,λ)=λw*(t,λ), ∂tnw*(t,λ)=−(∂−1(Ln)≥1*∂)(w*(t,λ)).(2.12)

It is proved in [3, 25] that there exist two tau functions τ₁ and τ₀ for the mKP hierarchy in Kupershmidt-Kiso version such that

w(t,λ)=τ0(t−[λ−1])τ1(t)eξ(t,λ),(2.13)

w*(t,λ)=τ1(t+[λ−1])τ0(t)λ−1e−ξ(t,λ).(2.14)

where [λ]=(λ,λ22,λ33,…). Then the bilinear identity (2.10) can be written into [3]

τ1(t′)τ0(t)=resλ(λ−1τ0(t′−[λ−1])τ1(t+[λ−1])eξ(t′−t,λ)).(2.15)

The k-constrained mKP hierarchy [3, 20] is defined by imposing the following constraints on the Lax operator,

Lk=(Lk)≥1+q∂−1r∂,(2.16)

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

qtn=(Ln)≥1(q), rtn=−(∂(Ln)≥1∂−1)*(r).(2.17)

That is to say, the k-constrained mKP hierarchy are the system of (2.2), (2.3), (2.16) and (2.17). Next we list the powers of the Lax operators and the flows for k = 1 and k = 2.

Case k = 1

Powers of the Lax operators (k = 1)
L=∂+qr−qrx∂−1+qrxx∂−2−qrxxx∂−3+⋯L2=∂2+2qr∂+(qxr−qrx+q2r2)+(qrxx−qxrx−2q2rrx)∂−1 +(qxrxx+2q2rx2+qqxrrx−qrxx+2q2rrxx)∂−2+⋯L3=∂3+3qr∂2+(3qxr+3q2r2)∂+qxxr+qrxx+3qqxr2−3q2rrx+q3r3−qxrx +(3q2rrxx+qxrxx−qrxxx−qxxrx−3qrqxrx−3q3r2rx+3q2rx2)∂−1+⋯(2.18)
Flows of the cmKP hierarchy (k = 1)
u0t2=qxxr+2qrrx−qrxx+2q2rrx,qt2=qxx+2qrqx, rt2=−rxx+2qrrx;(2.19)
and
u0t3=qxxxr+qrxxx+3r2(qqxx+(qx)2+q3rx) +3q2(r3qx−(rx)2+(qx)2)−3qqxrrx,qt3=qxxx+3qrqxx+3(qx)2r+3(qr)2qx,rt3=rxxx−3(qr)xrx+3(qx)2+3(qr)2rx.(2.20)

Case k = 2

Powers of the Lax operators (k = 2)
L=∂+u0+12(qr−u02−u0x)∂−1+14(−3qrx−qxr+4u0u0x+u0xx−2u0qr+2u03)∂−2+⋯L2=∂2+2u0∂+qr−qrx∂−1+qrxx∂−2−qrxxx∂−3+⋯L3=∂3+3u0∂2+32(u0x+u02+qr)∂+14(u0xx+3qxr−3qrx+6qru0−2u03) +(−14u0xxx+38q2r2+12u02u0x+54qrxx+14qxxr−34u0u0xx−54u0qrx +14u0qxr−34u02qr−u0xqr+18(u0x)2+38u04)∂−1+⋯(2.21)
Flows of the cmKP hierarchy (k = 2)
u0t2=(qr)x,qt2=qxx+2u0qx,rt2=−rxx+2u0rx;(2.22)
and
u0t3=32(qxxr+qxrx+u0(qr)x)+34(u0x+u02+qr)(1−qr)−u0(32u0xx+4u0u0x−u0x),qt3=qxxx+3u0qxx+32(u0x+u02+qr)qx,rt3=rxxx−3u0rxx+32(−u0x+u02+qr)rx.(2.23)

3. Bilinear identities

In this section, we would like to discuss the bilinear identity formulation of the k-constrained mKP hierarchy. From the spectrum equation of the linear system (2.11), it is obvious that

Lk(w(t,λ))=((Lk)≥1+q∂−1r∂)(w(t,λ))=λkw(t,λ),(3.1)

that is

(Lk)≥1(w(t,λ))+qS(t,λ)=λkw(t,λ),(3.2)

where

Sx(t,λ)=rwx(t,λ).(3.3)

Similarly, from (2.12), we have

∂−1(Lk)≥1*∂(w*(t,λ))+rS^(t,λ)=λkw*(t,λ),(3.4)

where

S^x(t,λ)=qwx*(t,λ).(3.5)

Before further dicussion, the following lemmas are needed.

Lemma 3.1 ([7]).

Let P and Q be two pseudo-differential operators, then

resλ[(Pexλ)(Qe−xλ)]=res∂PQ*,(3.6)

where Q^* is the formal adjoint of Q.

Lemma 3.2 ([7]).

If f(z)=∑i=0∞aiz−i is a standard series, one has the following operator identities:

resz((∑n=−∞∞an(ζ)z−n)11−z/ζ)=ζ(∑n=1∞an(ζ)z−n)|z=ζ.(3.7)

Proposition 3.1.

For the k-constrained mKP hierarchy (2.16), the eigenfunctions r(t) and q(t) satisfy the following residue formulas:

q(t)r(t′)=resλλkw(t,λ)w*(t′,λ),(3.8)

q(t)=resλw(t,λ)S^(t′,λ),(3.9)

r(t′)=resλS(t,λ)w*(t′,λ),(3.10)

with t and t′ being arbitrary and independent of each other.

Proof.

We first calculate the residue of L^k∂^m by Lemma 3.1 for an integer m ≥ −1

(−1)m+1q∂xm+1(r)=res∂(Lk∂m)=res∂(Z∂kZ−1∂m)=resλ(Z∂k(eξ(t,λ))⋅(−∂)m(Z*)−1(e−ξ(t,λ)))=(−1)m+1resλ(λkw(t,λ)⋅∂xm+1(w*(t,λ))).(3.11)

Then by using the Taylor expansion

f(t′)=∑(t1′−t1)i1⋯(tm′−tm)im∂1i1⋯∂mimf(t)/i1!⋯im!.(3.12)

and also using (2.12) and (2.17), one can obtain

q(t)r(t′)=resλλkw(t,λ)w*(t′,λ),(3.13)

Further from (2.10), we find that the formula (3.13) can be written as follows:

q(t)r(t′)=resλλkw(t,λ)w*(t′,λ)=resλw(t,λ)⋅(∂x′(Lk)≤0∂x′−1)*(w*(t′,λ))=resλw(t,λ)⋅r(t′)∂x′−1q(t′)∂x′(w*(t′,λ))=resλw(t,λ)r(t′)S^(t′,λ).(3.14)

By eliminating r(t′) on both sides of the upper form, we get (3.9). By the similar way, formula (3.13) can also be written the following form:

q(t)r(t′)=resλλkw(t,λ)w*(t′,λ)=resλLkw(t,λ)w*(t′,λ)=resλ(Lk)≤0w(t,λ)w*(t′,λ)=resλq(t)S(t,λ)w*(t′,λ).(3.15)

By eliminating q(t) from both sides of the equation above, (3.10) can be proved.

Proposition 3.2.

Conversely, let w(t, λ) be a formal power series of the form w(t,λ)=∑i=0∞ziλ−ieξ(t,λ), w*(t,λ)=(z0−1+∑i=0∞zi*λ−i)eξ(t,λ)λ−1, where z_i and z*i are functions of variable t_i. Both w(t, λ) and w^*(t, λ) satisfy (3.8)–(3.10). Then letting Z=∑i=0∞zi∂−i, one has the following conclusions:

(i)
w(t, λ) = Z(e^{ξ(t, λ)}), w^*(t, λ) = (Z⁻¹∂⁻¹)^* (e^{−ξ(t, λ)}),
(ii)
∂_{t_n}Z = (Z∂ⁿZ⁻¹)_≤0Z,
(iii)
(L^k)_≤0 = q∂⁻¹r∂.

Proof.

(i)–(ii) It is obvious that w(t, λ) = Z(e^{ξ(t, λ)}). Let

w*(t,λ)=−Z˜∂−1(e−ξ(t,λ)),(3.16)

where Z˜=z0−1+∑i=1∞zi*∂−1. By differentiating both sides of (3.9) with respect to x′ and letting t ↔ t′,

0=resλw(t′,λ)w*(t,λ)x.(3.17)

Then

0=resλ(z0−1(t′)Z)(eξ(t′,λ))⋅(z0(t)∂Z˜∂−1)(e−ξ(t,λ)).(3.18)

Note that the highest order terms of z0−1(t′)Z1 and z0(t)∂Z˜∂−1 in (3.18) are 1, which implys the residue formula (3.18) can be viewed as the bilinear relation of the KP hierarchy. Thus we have

z0−1Z⋅∂−1Z˜*∂z0=1(3.19)

and φ≜z0−1Z is the dressing operator of KP hierarchy, i.e. ϕ_{t_n} = −(ϕ∂ⁿϕ⁻¹)_<0ϕ. Therefore,

w*(t,λ)=−Z˜∂−1(e−ξ(t,λ))=(Z−1∂−1)*(e−ξ(t,λ)).(3.20)

If set ψ(t, λ) = ϕ(e^{ξ(t, λ)}), then ψ(t, λ)_{t_n} = B_n(ψ). Therefore

w(t,λ)tn=(z0ψ(t,λ))tn=z0tn⋅(z0−1Z)(eξ(t,λ))+z0⋅(Bnz0−1Z)(eξ(t,λ))=(z0tn⋅z0−1+z0⋅Bnz0−1)w(t,λ)).(3.21)

It can be found from equation (3.21) that the derivatives of w(t, λ) with respect to t_n can be transformed into the action of the differential operators ∑i≥0ai∂i on it. Then by Lemma 3.1

resλ∂iw(t,λ)w*(t,λ)=resλ∂iZ(eξ(t,λ))(Z−1∂−1)*(e−ξ(t,λ))=res∂∂iZ⋅Z−1∂−1=res∂∂i−1=δi0.(3.22)

Taking into account (3.12), one can obtain

resλw(t,λ)w*(t′,λ)x′=1,(3.23)

which is the bilinear identity of the mKP hierarchy, thus ∂_{t_n}Z = (Z∂ⁿZ⁻¹)_≤0Z.

(iii) Differentiating (3.8) with respect to t′_n, one sees that

q(t)∂n′r(t′)=−resλλkw(t,λ)(∂x′−1(Ln)≥1*∂x′)(w*(t′,λ))=−(∂x′−1(Ln)≥1*∂x′)(q(t)r(t′))=−q(t)(∂x′−1(Ln)≥1*∂x′)(r(t′)),(3.24)

which means that r(t) is the adjoint eigenfunction. By the similar way, one show that q(t) is the eigenfunction. Finally, by differentiating (3.8) with respect to t′_n and letting t′ = t,

q(t)∂nr(t)=resλλkw(t,λ)⋅∂n(w*(t′,λ))=resλ(LkZ)(eλx)⋅∂n(Z−1∂−1)*(e−λx)=(−1)nres∂Lk∂n−1,(3.25)

that is,

(−1)nq(t)∂n(r(t))=res∂Lk∂n−1.(3.26)

Formula (3.26) shows that the ∂⁻ⁿ term in L^k is (−1)ⁿq(t)r⁽ⁿ⁾(t). Furthermore, ∑n=1∞(−1)nqr(n)∂−n=q∂−1r∂. This means that the nonpositive part of the Lax operator L^k is the form (2.16). This completes the proof that (3.8), (3.9) and (3.10) fully characterize the k-constrained mKP hierarchy.

4. Tau functions and Hirota bilinear equations

In this section, we would like to rewrite the bilinear identities in Hirota form. Introduce two auxiliary functions ρ(t) and σ(t) such that

q(t)=ρ(t)τ1(t), r(t)=σ(t)τ0(t),(4.1)

where τ₀(t) and τ₁(t) are defined by (2.13) and (2.14), then we have the explicit expressions for functions S(t, λ) and Ŝ(t, λ).

Proposition 4.1.

S(t,λ)=σ(t−[λ−1])τ1(t)eξ(t,λ),(4.2)

S^(t,λ)=ρ(t+[λ−1])λτ0(t)e−ξ(t,λ).(4.3)

Proof.

In terms of the definition of S(t, λ) (see (3.3)), the function S can be expressed as

S(t,λ)=K1(t,λ)eξ(t,λ),(4.4)

where

K1(t,λ)=k0+k1λ+k2λ2+⋯.(4.5)

Due to

1(1−zs1)(1−zs2)=(1(1−zs2)−1(1−zs1))1z(s2−s1).(4.6)

Define the operator G(z) f(t) = f(t − [z⁻¹]). By setting the time variables tn′=tn−1nξ1n−1nξ2n , and substituting (4.4) and (2.8) into (3.10), and using Lemma 3.2,

G(ξ1)G(ξ2)r(t)=resλ(K1(t,λ)eξ(t,λ)w^*(t′,λ)λ−1e−ξ(t′,λ))=resλ(K1(t,λ)eξ(t,λ)G(ξ1)G(ξ2)w^*(t,λ)λ−1e−ξ(t,λ)(1−λξ1)(1−λξ2))=ξ11−ξ1/ξ2(K1(t,ξ1)G(ξ1)G(ξ2)w^*(t,ξ1)1ξ1−K1(t,ξ2)G(ξ1)G(ξ2)w^*(t,ξ2)1ξ2).

By setting ξ₁ = λ, ξ₂ → ∞, the above equation reads

G(λ)σ(t)G(λ)τ0(t)=K1(t,λ)τ1(t)G(λ)τ0(t).(4.7)

We then get the expression of S(t, λ) as follows

S(t,λ)=K1(t,λ)eξ(t,λ)=G(λ)σ(t)G(λ)τ0(t)⋅G(λ)τ0(t)τ1(t)=σ(t−[λ−1])τ1(t)eξ(t,λ).(4.8)

In a similar way, from the definition of Ŝ(t, λ), the function Ŝ can be expressed as

S^(t,λ)=K2(t,λ)e−ξ(t,λ)=(k¯1λ+k¯2λ2+⋯)e−ξ(t,λ).(4.9)

By substituting (4.9) and (2.7) into the bilinear identity (3.9) and taking tn=tn′+1nz1n+1nz2n,

G(−z1)G(−z2)q(t′)=resλ(G(−z1)G(−z2)w^(t′,λ)eξ(t′,λ)+ξ(z1,λ)+ξ(z2,λ)K2(t′,λ)e−ξ(t′,λ))=resλ(K2(t′,λ)G(−z1)G(−z2)w^(t′,λ)(1−λz1)(1−λz2))=z11−z1/z2(K2(t′,z1)G(−z1)G(−z2)w^(t′,z1)−K2(t′,z2)G(−z1)G(−z2)^^w^(t′,z2)).

By setting z₁ = λ, z2−1=0, we finally reach (t ↔ t′)

G(−λ)ρ(t)λG(−λ)τ1(t)=K2(t,λ)τ0(t)G(−λ)τ1(t),(4.10)

which immediately implies

S^(t,λ)=ρ(t+[λ−1])λτ0(t)e−ξ(t,λ).(4.11)

Proposition 4.2.

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

ρ(t)σ(t′)=resλ(λk−1τ0(t−[λ−1])τ1(t′+[λ−1])eξ(t−t′,λ))(4.12)

ρ(t)τ0(t′)=resλ(λ−1τ0(t−[λ−1])ρ(t′+[λ−1])eξ(t−t′,λ))(4.13)

σ(t′)τ1(t)=resλ(λ−1τ1(t′−[λ−1])σ(t−[λ−1])eξ(t−t′,λ))(4.14)

Proof.

These equations can be proved by substituting (2.13), (2.14), (4.1)–(4.3) into (3.8)–(3.10).

Remark 4.1.

From (4.13) and (4.14), one can find that (τ₀, ρ) and (σ, τ₁) can be viewed as the τ functions of the mKP hierarchy, since they share the same form with (2.15).

Next, we try to rewrite (4.12)–(4.14) into the Hirota forms. For a polynomial P, one can define the Hirota bilinear operator [9] as follows:

P(D)f(t)⋅g(t)=((∂∂t1−∂∂t1′),(∂∂t2−∂∂t2′),…)(f(t)g(t′))|t=t′=P(∂y)(f(t+y)g(t−y))|y=0.(4.15)

Here P(D) = P(D₁, D₂,...) and ∂_y = (∂_y₁, ∂_y₂,...). Another important object is the Schur polynomials p_n(t), which are defined in the following way,

eξ(t,λ)=∑n=0∞pn(t)λn.(4.16)

One can find p_n(t) owns the form below:

pn(t)=∑‖α‖=ntαα!,(4.17)

where α = (α₁, α₂,...), α_j ≥ 0, ‖α‖=∑j=0∞jαj, α! = α₁!α₂!··· , tα=t1α1t2α2⋯.

Proposition 4.3.

The Hirota’s bilinear forms of the cmKP hierarchy are

(1) ∑α+β=γ(−2)αα!β!pk+‖α‖(D˜)Dβτ1(t)⋅τ0(t)−1γ!Dγσ(t)⋅ρ(t)=0,(4.18)

(2) (∑α+β=γ(−2)αα!β!p‖α‖(D˜)Dβ−(−1)|γ|γ!Dγ)ρ(t)⋅τ0(t)=0,(4.19)

(3) (∑α+β=γ(−2)αα!β!p‖α‖(D˜)Dβ−(−1)|γ|γ!Dγ)τ1(t)⋅σ(t)=0.(4.20)

where D˜=(D1,D22,D33,…).

Proof.

(1) Changing the variables in (4.12), t → t − y, t′ → t + y,

ρ(t−y)σ(t+y)=resλ(λk−1τ0(t−y−[λ−1])τ1(t+y+[λ−1])eξ(−2y,λ))=resλ(λk−1eξ(−2y,λ)e∑n=1∞(1nλn)∂yn(τ0(t−y)τ1(t+y)))=resλ(∑j=0∞pj(−2y)λk+j−1∑n=0∞pn(∂˜y)λ−n(τ0(t−y)τ1(t+y)))=∑j=0∞pj(−2y)pk+j(∂˜y)(τ0(t−y)τ1(t+y)).(4.21)

By considering that one can expand f(y) = f(y₁, y₂,...) at y = 0 in the following way

f(y)=exp(∑i=1∞yi∂∂zi)f(z)|z=0,(4.22)

one can further obtain

exp(∑i=1∞yi∂∂zi)(ρ(t−z)σ(t+z))|z=0=exp(∑i=1∞yi∂∂zi)∑j=0∞pj(−2y)⋅pj(∂˜z)(τ0(t−z)τ1(t+z))|z=0,(4.23)

where ∂˜z=(∂z1,12∂z2,13∂z3,…). Therefore

e∑i=0∞yiDiσ(t)⋅ρ(t)=∑j=0∞pj(−2y)pk+j(D˜)e∑n=1∞ynDnτ1(t)⋅τ0(t).(4.24)

Next, one can rewrite (4.24) into a more explicit form. Before doing this, the next formula is needed.

exp(∑j=0∞yj)=∑|α|=0∞yαα!,(4.25)

where y = (y₁, y₂,...), |α|=∑k=1∞αk.

By (4.17) and (4.25), we can rewrite (4.24) as

∑|β|=0∞yββ!Dβ(σ(t)⋅ρ(t))=∑j=0∞∑‖α‖=j(−2y)αα!pk+j(D˜)∑|β|=0yββ!Dβτ1(t)⋅τ0(t)=∑‖α‖=0,|β|=0(−2y)αα!pk+‖α‖(D˜)yββ!Dβτ1(t)⋅τ0(t).(4.26)

Further,

∑γ(∑α+β=γ(−2)αα!β!pk+‖α‖(D˜)Dβτ1(t)⋅τ0(t)−1γ!Dγσ(t)⋅ρ(t))yγ=0(4.27)

Therefore, one can obtain (4.18). In the same way, it is easy to prove (4.19) and (4.20).

Equations in (4.18), (4.19) and (4.20) give rise to the hierarchy of Hirota bilinear equations corresponding to the k-constrained mKP hierarchy. Let us show some examples for k = 1 and k = 2.

Case k = 1

For γ = 0,
σ⋅ρ=D1τ1⋅τ0.(4.28)
For γ = (1, 0, 0,...),
D1σ⋅ρ=−2D2τ1⋅τ0.(4.29)
For γ = (0, 1, 0,...),
D2σ⋅ρ+(D1D2+13D13+2D3)τ1⋅τ0=0,(D2−D12)ρ⋅τ0=0, (D2+D12)σ⋅τ1=0.(4.30)
For γ = (0, 0, 1,...),
D3σ⋅ρ+(2D4+D1D3+D22+112D14+D12D2)τ1⋅τ0=0,(4D3−3D13−3D1D2)ρ⋅τ0=0,(4D3−3D13+3D1D2)τ1⋅σ=0.(4.31)

Case k = 2

For γ = 0,
2σ⋅ρ=(D12+D2)τ1⋅τ0.(4.32)
For γ = (1, 0, 0,...),
D1σ⋅ρ+(12D1D2+23D3+12D13)τ1⋅τ0=0.(4.33)
For γ = (0, 1, 0,...),
D2σ⋅ρ+(112D14+14D22+23D1D3+12D4)τ1⋅τ0=0,(D2−D12)ρ⋅τ0=0, (D2+D12)σ⋅τ1=0.(4.34)
For γ = (0, 0, 1,...),
D3σ⋅ρ+(2D5+D22+52D1D4+32D12D3+12D12D2+14D1D22+12D13D2+512D15)τ1⋅τ0=0,(4D3−3D13−3D1D2)ρ⋅τ0=0,(4D3−3D13+3D1D2)τ1⋅σ=0.(4.35)

At last, we will use the gauge transformation to give the specific form of ρ(t) and σ(t). In [1], we have constructed the gauge transformation of the constrained mKP hierarchy. For the cmKP hierarchy (L(j))≤0k=q(j)∂−1r(j)∂, using the n steps of gauge transformation operator T_D,

L(0)→TD(q(0))L(1)→TD(q(1))L(2)→TD(q(2))…→TD(n−1)L(n),

we have

q(n)=(−1)nWn+1(q(0),η1,η2,…,ηn)Wn(q(0),η1,η2,…,ηn−1),r(n)=(−1)n−1Wn+1(q(0),η1,η2,…,ηn−2)Wn(q(0),η1,η2,…,ηn−1)τ0(n)=Wn(q(0),η1,…,ηn−1)(τ1(0))n(τ0(0))n−1,τ1(n)=Wn+1(q(0),η1,…,ηn−1,1)(τ1(0))n+1(τ0(0))n,(4.36)

where

TD(q(j))=((q(j))−1)x−1∂(q(j))−1, ηj=(L(0))kj(q(0)).(4.37)

Here we have used the results in [2] about the changes of the tau functions under the gauge transformations.

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

L(0)→TI(r(0))L(1)→TI(r(1))L(2)→TI(r(2))…→TI(r(n−1))L(n),

and

r(n)=Wn+1(r(0),η^1,η^2,…,η^n)Wn+1(1,r(0),η^1,η^2,…,η^n−1),q(n)=Wn(1,r(0),η^1,η^2,…,η^n−2)Wn(r(0),η^1,η^2,…,η^n−1),τ0(n)=Wn+1(1,r(0),η1,…,ηn−1)(τ1(0))n(τ0(0))n−1,τ0(n)=Wn+1(1,r(0),η^1,…,η^n−1)(τ0(0))n+1(τ1(0))n,τ1(n)=Wn(r(0),η^1,…,η^n−1)(τ0(0))n(τ1(0))n−1,(4.38)

where

TI(r(j))=(r(j))−1∂−1(r(j))x, η^j=∫(L(0)*)kj(rx(0))dx.(4.39)

Also the transformed tau functions can be derived by using the results in [2].

Starting from the zero solution of the mKP hierarchy, i.e. L⁽⁰⁾ = ∂, τ0(0)=τ1(0)=1, we have the following results.

Proposition 4.4.

Under the gauge transformation operator T_D(q),

ρ(n)(t)=(−1)nWn+1(q(0),η1,…ηn)Wn+1(q(0),η1,…ηn−1,1)Wn(q(0),η1,…ηn−1),(4.40)

σ(n)(t)=(−1)n−1Wn−1(q(0),η1,…ηn−2).(4.41)

Under the gauge transformation operator T_I(r),

ρ(n)(t)=Wn(1,r(0),η^1,…,η^n−2)(4.42)

σ(n)(t)=Wn+1(r(0),η^1,…,η^n).(4.43)

5. Conclusions and Discussions

The main results of this paper are as follows. We firstly derive the bilinear identities of the constrained mKP hierarchy from the calculus of the pseudo-differential operators, which are summarized in Proposition 3.1 and Proposition 3.2. In order to write bilinear equations in the form of Hirota operators, we introduce the auxiliary functions ρ and σ in Section 4, and give their bilinear equations in Proposition 4.2. Then, the Hirota’s bilinear forms of the cmKP hierarchy are given in Proposition 4.3. At last, we use the gauge transformation to give the specific form of ρ(t) and σ(t) in Proposition 4.4.

We have established the bilinear method to express the constrained mKP hierarchy, just from the constraint on the Lax operator and the evolution equations of the (adjoint) wave functions. Though there is the Miura link between the KP and mKP hierarchy [22], our results are still not obvious. Comparatively, it is more difficult to obtain the bilinear formulation, only by using the Miura link from the results in KP case. Another important point is the auxiliary functions ρ and σ. Since (τ₀, ρ) and (σ, τ₁) can be viewed as the new tau functions of the mKP hierarchy, it will be very interesting to further understand ρ and σ.

Acknowledgments

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

References

[1]H.Z. Chen, L.M. Geng, N. Li, and J.P. Cheng, Solving the constrained modified KP hierarchy by gauge transformations, J. Nonlin. Math Phys., Vol. 26, 2019, pp. 1-15.

[2]J.P. Cheng, The gauge transformation of modified KP hierarchy, J. Nonlin. Math. Phys., Vol. 25, 2018, pp. 66-85.

[3]J.P. Cheng, M.H. Li, and K.L. Tian, On the modified KP hierarchy: tau functions, squared, eigenfunction symmetries and additional symmetries, J. Geom. Phys., Vol. 134, 2018, pp. 19-37.

[4]Y. Cheng, Modifying the KP, the nth constrained KP hierarchies and their Hamiltonian structures, Comm. Math. Phys., Vol. 171, 1995, pp. 661-682.

[5]Y. Cheng and Y.J. Zhang, Bilinear equations for the constrained KP hierarchy, Inverse Problems, Vol. 10, 1994, pp. L11-L17.

[6]L.A. Dickey, Modified KP and discrete KP, Lett. Math. Phys., Vol. 48, 1999, pp. 277-289.

[7]L.A. Dickey, Soliton equations and Hamiltonian systems, second ed., World scientific, Singapore, 2003.

[8]A. Doliwa and R.L. Lin, Discrete KP equation with self-consistent sources, Phys. Lett. A, Vol. 378, 2014, pp. 1925-1931.

[9]R. Hirota, The Direct method in soliton theory, Cambridge University Press, 2004.

[10]M. Jimbo and T. Miwa, Solitons and infinite dimensional Lie algebras, Publ. RIMS, Kyoto Univ., Vol. 19, 1983, pp. 943-1001.

[11]V. Kac and J. van de Leur, Equivalence of formulations of the mKP hierarchy and its polynomial tau-functions, Japan. J. Math., Vol. 13, 2018, pp. 235-271.

[12]M. Kashiwara and T. Miwa, The τ function of the Kadomtsev-Petviashvili equation, Transfromation groups for soliton equations, I, Proc. Japan Acad. A, Vol. 57, 1981, pp. 342-347.

[13]K. Kiso, A remark on the commuting flows defined by Lax equations, Prog. Theor. Phys., Vol. 83, 1990, pp. 1108-1125.

[14]B. Konopelchenko and W. Oevel, An r-matrix approach to nonstandard classes of integrable equations, Publ. RIMS Kyoto Univ., Vol. 29, 1993, pp. 581-666.

[15]B. A. Kupershmidt, Mathematics of dispersive water waves, Comm. Math. Phys., Vol. 99, 1985, pp. 51-73.

[16]C.Z. Li, J.S. He, Ke Wu, and Yi Cheng, Tau function and Hirota bilinear equations for the extended bigraded Toda hierarchy, J. Math. Phys., Vol. 51, 2010, pp. 043514.

[17]I. Loris and R. Willox, Bilinear form and solutions of the k-constrained Kadomtsev-Petviashvili hierarchy, Inverse Problems, Vol. 13, 1997, pp. 411-420.

[18]T. Milanov, Hirota quadratic equations for the extended Toda hierarchy, Duke Math. J., Vol. 138, 2007, pp. 161-178.

[19]W. Oevel and S. Carillo, Squared eigenfunction symmetries for soliton equations: Part I, J. Math. Anal. Appl., Vol. 217, 1998, pp. 161-178.

[20]W. Oevel and S. Carillo, Squared eigenfunction symmetries for soliton equations: Part II, J. Math. Anal. Appl., Vol. 217, 1998, pp. 179-199.

[21]W. Oevel and C. Rogers, Gauge transformations and reciprocal links in 2 + 1 dimensions, Rev. Math. Phys., Vol. 5, 1993, pp. 299-330.

[22]J.C. Shaw and M.H. Tu, Miura and auto-Backlund transformations for the cKP and cmKP hiearchies, J. Math. Phys., Vol. 38, 1997, pp. 5756-5773.

[23]H.F. Shen, N.C. Lee, and M.H. Tu, Operator approach to solutions of the constrained BKP hierarchy, J. Phys. A: Math. Theor., Vol. 44, 2011, pp. 135205.

[24]H.F. Shen and M.H. Tu, On the constrained B-type Kadomtsev-Petviashvili hierarchy: Hirota bilinear equations and Virasoro symmetry, J. Math. Phys., Vol. 52, 2011, pp. 032704.

[25]T. Takebe and L.P. Teo, Coupled modified KP hierarchy and its dispersionless limit, SIGMA, Vol. 2, 2006, pp. 072.

[26]Y.Q. Yao, J.H. Zhang, R.L. Lin, X.J. Liu, and Y. H. Huang, Bilinear identities and Hirota’s bilinear forms for the (γn, σk)-KP hierarchy, J. Nonlin Math. Phys., Vol. 25, 2018, pp. 309-323.

<Previous Article In Issue

Download article (PDF)

Next Article In Issue>

Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 26 - 2
Pages: 240 - 254
Publication Date: 2021/01/06
ISSN (Online): 1776-0852
ISSN (Print): 1402-9251
DOI: 10.1080/14029251.2019.1591721 How to use a DOI?
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

ris enw bib

TY  - JOUR
AU  - Huizhan Chen
AU  - Lumin Geng
AU  - Na Li
AU  - Jipeng Cheng
PY  - 2021
DA  - 2021/01/06
TI  - Bilinear identities for the constrained modified KP hierarchy
JO  - Journal of Nonlinear Mathematical Physics
SP  - 240
EP  - 254
VL  - 26
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2019.1591721
DO  - 10.1080/14029251.2019.1591721
ID  - Chen2021
ER  -

download .riscopy to clipboard