Two-component generalizations of the Novikov equation

Hongmin Li

doi:10.1080/14029251.2019.1613048

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Volume 26, Issue 3, May 2019, Pages 390 - 403

Two-component generalizations of the Novikov equation

Authors

Hongmin Li

Schools of Mathematics, Huaqiao University, Quanzhou, Fujian 362021, P R China,lihongmin@hqu.edu.cn

Received 1 November 2018, Accepted 2 March 2019, Available Online 6 January 2021.

DOI: 10.1080/14029251.2019.1613048 How to use a DOI?
Keywords: Camassa-Holm type equation; Lax pair; Bi-Hamiltonian structure
Abstract: Some two-component generalizations of the Novikov equation, except the Geng-Xue equation, are presented, as well as their Lax pairs and bi-Hamiltonian structures. Furthermore, we study the Hamiltonians of the Geng-Xue equation and discuss the homogeneous and local properties of them.
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 recent decades, the Camassa-Holm (CH) type equations raised a lot of interest because of their specific properties, one of which is that they possess peakon solutions (peaked soliton solutions with discontinuous derivatives at the peaks). The most celebrated member of them is the CH equation [1]

mt+umx+2uxm=0, m=u−uxx,(1.1)

which was found to be completely integrable with a Lax pair and associated bi-Hamiltonian structure [1, 2], and related by a reciprocal transformation to the first negative flow in the KdV hierarchy [18]. It is worthwhile to note that an inverse scattering approach for the CH equation was developed in the Refs. [5, 11, 14], which showed that after a suitable change of variables, the CH-flow becomes a linear flow at constant speed, this being the strong version of integrability (the infinite-dimensional counterpart of the classical Liouville theorem). Moreover, while an initial m without change of sign leads to solutions of CH that are defined for all times t ≥ 0 [3,8], among the initial m that change sign one encounters solutions of CH that model breaking waves, in the sense that the solution itself remains bounded but its slope becomes unbounded [10].

In 1999, Degasperis and Procesi used an asymptotic integrability approach to isolate integrable third-order equations, and discovered a new CH type equation, i.e., the Degasperis-Procesi (DP) equation

mt+umx+3uxm=0, m=u−uxx,(1.2)

which turns out to be integrable with a bi-Hamiltonian structure and a Lax pair [16,17], and is reciprocal linked to a negative flow in the Kaup-Kupershmidt hierarchy [22]. Furthermore, the inverse scattering problem for the DP equation was formulated in Ref. [12]. Especially, for CH and DP equations one can prove that the peakons are orbitally stable [15, 25, 31], in the sense that their shape is stable under small perturbations, and therefore these patterns are detectable. This is an essential aspect since both equations arise as models for water waves [13, 23, 24]. Note that the travelling wave solutions of greatest height of the governing equations for water waves have a peak at their crest [6, 7, 9, 35]. The proof of the orbital stability of the peakons relies on the conservation laws for CH and DP.

Both the CH and DP equations are the third order CH type equations with quadratic nonlinearity. Recently, by the symmetry classification study of nonlocal partial differential equations with quadratic or cubic nonlinearity, Novikov discovered a new CH type equation with cubic nonlinearity [32]

mt+u2mx+3uuxm=0, m=u−uxx.(1.3)

Subsequently, Hone and Wang presented a Lax pair for the Novikov equation (1.3) and proved it is related to the negative flow in the Sawada-Kotera hierarchy by a reciprocal transformation. Infinitely many conserved quantities and a bi-Hamiltonian structure of the Novikov equation are also constructed [21].

Later Geng and Xue [19] presented a two-component generalization for the Novikov equation

mt+3uxvm+uvmx=0,nt+3vxun+uvnx=0,m=u−uxx,n=v−vxx,(1.4)

which admits a Lax representation

φx=M¯φ, φt=N¯φ,(1.5)

where

M¯=(0mλ100nλ100),N¯=(13λ2−uxvuxλ−uvmλuxvxvλ−23λ2+uxv−uvx−uvnλ−vxλ−uvuλ13λ2+uvx).(1.6)

They also gave an infinite sequence of conserved quantities of the system (1.4) (also called Geng-Xue equation). Li et al proved it is bi-Hamiltonian [29] and reciprocal linked to a negative flow in the modified Boussinesq hierarchy [30]. Very recently, we constructed a Liouville transformation to connect it with another negative modified Boussinesq equation, and Lax pairs as well as bi-Hamiltonian structures of them are connected [27].

Recently, we make the vector prolongation of the Lax pair (1.5) as follows [28]

Φx=M˜Φ, Φt=N˜Φ,(1.7)

with

M˜=(0λQT10T0λR100),N˜=(−UxTVUxTλ−λUTVQUxTVxVλ−INλ2+VUxT−VxUT−Vxλ−λUTVR−UTVUTλUTVx),(1.8)

where 0 and 0_N are respectively N dimension row vector and N × N zero matrix and T is the transpose of a vector. I_N denotes the N × N identity matrix and Q, R, U, V are the N-component column vector potentials, and UxT=∂UT∂x.

Then the zero-curvature equation for (1.7) yields the multi-component Novikov equation

Qt=−2UxTVQ−UTVxQ−UTVQx−QTVUx+QTVxU,Rt=−2UTVxR−UxTVR−UTVRx−RTUVx+RTUxV,Q=U−Uxxx, R−V−Vxx,(1.9)

which is bi-Hamiltonian [28] proved by the multi-vector method [34]. This multi-component equation can be reduced to Geng-Xue equation (1.4), DP equation (1.2) and Novikov equation (1.3) under the constraints N = 1, Q = m, R = n, U = u, V = v; N = 1, Q = m, U = u, R = V = 1 and N = 1, Q = R = m, V = U = u respectively. The aim of this paper is to consider some reductions of the Lax pair (1.7) and the corresponding evolution equations.

This paper is organized as follows. In Section 2, we consider two interesting reductions of the multi-component Novikov equation (1.9) and construct the bi-Hamiltonian structures for them. In Section 3, we compute two infinite sequences of conserved quantities for the Geng-Xue equation, and discuss the homogeneous and local properties of the Hamiltonians of the Geng-Xue equation.

2. Two-component generalizations of the Novikov equation

In this section, we will study two reductions of the multi-component Novikov equation (1.9) and obtain two reduced systems, as well as their bi-Hamiltonian structures using Dirac reduction [33].

2.1. The first two-component generalization of the Novikov equation

Setting U = V, Q = R, the multi-component Novikov equation (1.9) is reduced to

Qt=−3UxTUQ−UTUQx−QTUUx+QTUxU, Q=U−Uxx,(2.1)

which can be transformed to a two-component Novikov equation

qt=−(u2+v2)qx−3(uux+vvx)q+r(uvx−uxv),rt=−(u2+v2)rx−3(uux+vvx)r+q(uxv−uvx),q=u−uxx, r=v−vxx.(2.2)

as Q = R = (q, r)^T, U = V = (u,v)^T. Especially when u = v, the equation (2.2) is reduced to the Novikov equation (1.3).

As pointed in [28], the equation (1.9) can be written in the bi-Hamiltonian form

(QR)t=𝒦(δH2δQδH2δR)=𝒥(δH1δQδH1δR),(2.3)

using the two compatible Hamiltonian operators

𝒦=(0N(∂2−1)IN(1−∂2)IN0N),(2.4)

𝒥=𝒥1+𝒥2,(2.5)

with

𝒥1=(32Q∂+Qx32R∂+Rx)(∂3−4∂)−1(3QT∂+QxT3RT∂+RxT),𝒥2=(12Q∂−1QT+(Q∂−1QT)T−12Q∂−1RT−QT∂−1RIN−12R∂−1QT−RT∂−1QIN12R∂−1RT+(R∂−1RT)T),

and the two Hamiltonian functionals

ℋ1=12∫QTV+RTUdx,ℋ2=12∫QTVUxTV−RTUVxTU+(RTUx−QTVx)UTVdx.(2.6)

In what follows we will construct the bi-Hamiltonian structure for the multi-component Novikov equation (2.1) from (2.3). Introducing P=Q+R2, S=Q−R2, the new system with the variables P, S has a Hamiltonian operator:

∏(P,S)=14(INININ−IN)𝒥(INININ−IN),

then after using the Dirac reduction [33] under the constraint Q = R or S = 0, we have

Π(P,0)=12(Π11(P,0)Π12(P,0)Π21(P,0)Π22(P,0)),

where Π11)=(3Q∂+2Qx)(∂3−4∂)−1(3QT∂+Q xT)+(Q∂−1QT)T−QT∂−1QIN, Π₁₂(P,0) = Π₂₁(P,0) = 0, Π₂₂(P,0) = Q∂⁻¹Q^T + (Q∂⁻¹Q^T)^T + Q^T ∂⁻¹QI_N.

So one may obtain the reduced Hamiltonian operator for the multi-component equation (2.1) as follows:

Πred(Q)=12(Π11(P,0)−Π12(P,0)[Π22(P,0)]−1Π21(P,0))=12Π01(P,0).(2.7)

To obtain the second Hamiltonian operator for the multi-component equation (2.1), we consider the Hamiltonian operator 𝒦 𝒥⁻¹𝒦 and its Dirac reduction, then the following equality holds:

Λ(P,S)=14(INININ−IN)𝒦𝒥−1𝒦(INININ−IN).

Similarly, after the Dirac reduction, one may get

Λ(P,0)=12(∂2−1)(Λ11(P,0)Λ12(P,0)Λ21(P,0)Λ22(P,0))−1(∂2−1),

with Λ₁₁(P,0) = Π₂₂(P,0), Λ₁₂(P,0) = Λ₂₁(P,0) = 0 and Λ₂₂(P,0) = Π₁₁(P,0). So we have

Λ(P,0)=−12(∂2−1)([Π22(P,0)]−100[Π11(P,0)]−1)(∂2−1),

which means the second Hamiltonian operator for the equation (2.1) is

Λred(Q)=−12(∂2−1)[ Π22(P,0) ]−1(∂2−1).(2.8)

Specially, from the constraint Q = R = (q, r)^T, U = V = (u,v)^T and the Hamiltonian pair (2.7–2.8), the two-component system (2.2) possesses the bi-Hamiltonian operators

Πred(q,r)=(3q∂+2qr3r∂+2rx)(∂3−4∂)−1(3q∂+qx,3r∂+rx) +(r−q)∂−1(−r,q),(2.9)

Λred(q,r)=−12(∂2−1)(3q∂−1q+r∂−1rq∂−1r+r∂−1qq∂−1r+r∂−1qq∂−1q+3r∂−1r)−1(∂2−1).

Remark 2.1.

The two-component system (2.2) appears in the bi-Hamiltonian form

(qr)t=Πred(q,r)(δHδqδHδr)=Λred(q,r)(δWδqδWδr),(2.10)

with Hamiltonian pair Π^red(q, r), Λ^red(q, r) given by (2.9). The associated Hamiltonian functional is H=12∫qu+rvdx, and W is nonlocal and looks very complicated, so we omit it.

2.2. The second two-component generalization of the Novikov equation

Assuming N is an even number, we consider the constraint

Q=(Q1Q2),R=(R1R2),U=(U1U2),V=(V1V2),(2.11)

where R₁ = Q₂, R₂ = Q₁, V₁ = U₂, V₂ = U₁ and all of them are N2 dimension column vectors, then the multi-component Novikov equation (1.9) reduces to

Q1t=−3(U1TU2)xQ1−2U1TU2Q1x+Q1T(U2xU1−U2U1x) +Q2T(U1xU1−U1U1x),Q2t=−3(U1TU2)xQ2−2U1TU2Q2x+Q1T(U2xU2−U2U2x) +Q2T(U1xU2−U1U2x),Q1=U1−U1xxQ2=U2−U2xx.(2.12)

In particular, setting Q = (q, r)^T, R = (r, q)^T, U = (u, v)^T, V = (v, u)^T, the multi-component equation (2.12) leads to another two-component Novikov equation

qt=−2uvqx−2uvxq−4uxvq,rt=−2uvrx−2uxvr−4uvxr,q=u−uxx, r=v−vxx,(2.13)

which may be reduced to the Novikov equation (1.3) under the constraint u = v.

To construct the bi-Hamiltonian structure of the multi-component system (2.12), we apply the same procedure in subsection 2.1 and introduce these symbols P1=Q1+R22, P2=Q2+R12, S1=Q1−R22, S2=Q2−R12, then the Hamiltonian operator Π(P₁, P₂, S₁, S₂) for the new system with the variables P₁, P₂, S₁, S₂ can be written as:

∏(P1,P2,S1,S2)=14ℱ𝒥ℱ*,

where ℱ=(I00I0II0I00−I0I−I0), 0 and I are respectively N2×N2 zero matrix and N2×N2 identity matrix.

After using the Dirac reduction under the constraint Q₁ = R₂, Q₂ = R₁ or S₁ = S₂ = 0, the operator Π(P₁, P₂, S₁, S₂) reduces to

Π(P1,P2,0,0)=(Π11(P1,P2,0,0)Π12(P1,P2,0,0)00Π21(P1,P2,0,0)Π22(P1,P2,0,0)0000Π33(P1,P2,0,0)Π34(P1,P2,0,0)00Π43(P1,P2,0,0)Π44(P1,P2,0,0))

with

Π11(P1,P2,0,0)=12(3Q1∂+2Q1x)(∂3−4∂)−1(3Q 1T∂+Q 1xT)+12(Q1∂−1Q 1T)T,Π12 −12(Q 1T∂−1Q2+Q 2T∂−1Q1)I.Π21(P1,P2,0,0)=12(3Q2∂+2Q2x)(∂3−4∂)−1(3Q 1T∂+Q 1xT)+12(Q1∂−1Q 2T)T −12(Q 1T∂−1Q2+Q 2T∂−1Q1)I,Π22(P1,P2,0,0)=12(3Q2∂+2Q2x)(∂3−4∂)−1(3Q 2T∂+Q 2xT)+12(Q2∂−1Q 2T)T,Π33(P1,P2,0,0)=12(Q1∂−1Q 1T+(Q1∂−1Q 1T)T,Π34(P1,P2,0,0)=12(Q1∂−1Q 2T+(Q2∂−1Q 1T)T)+12(Q 1T∂−1Q2+Q 2T∂−1Q1)I,Π43(P1,P2,0,0)=12(Q2∂−1Q 1T+(Q1∂−1Q 2T)T)+12(Q 2T∂−1Q1+Q 1T∂−1Q2)I,Π44(P1,P2,0,0)=12(Q2∂−1Q 2T+(Q2∂−1Q 2T)T).

And then the reduced Hamiltonian operator for the multi-component Novikov equation (2.12) is

Πred(Q1,Q2)=(Π11(P1,P2,0,0)Π12(P1,P2,0,0)Π21(P1,P2,0,0)Π22(P1,P2,0,0)).

Moreover, by studying the Hamiltonian operator 𝒦 𝒥⁻¹𝒦 and its Dirac reduction, the second Hamiltonian operator for the multi-component system (2.12) may be obtained as

Λred(Q1,Q2)=−14(∂2−1)(Π44(P1,P2,0,0)Π43(P1,P2,0,0)Π34(P1,P2,0,0)Π33(P1,P2,0,0))−1(∂2−1).

Especially under the constraint Q = (q, r)^T, R = (r, q)^T, U = (u, v)^T, V = (v, u)^T, the two-component system (2.13) has a Hamiltonian pair

Πred(q,r)=12(3q∂+2qx3r∂+2rx)(∂3−4∂)−1(3q∂+qx3r∂+rx) +12(q∂−1q−q∂−1r−r∂−1qr∂−1r),Λred(q,r)=−14(∂2−1)(r∂−1rq∂−1+r∂−1qq∂−1r+r∂−1qq∂−1q)(∂2−1)=12(∂2−1)(−1q∂qr∂−1qr∂1q1q(2−(qr)x∂−1rq)∂1r1r(2−(rr)x∂−1qr)∂1q−1r∂rq∂−1rq∂1r)(∂2−1).(2.14)

Remark 2.2.

The two-component system (2.13) can be written in bi-Hamiltonian form (2.10) with bi-Hamiltonian operators given by (2.14). The associated Hamiltonian functional for Π^red(q, r) is H = ∫qv+rudx. The omission of W here is caused by the nonlocal property and complication of it.

3. Conserved quantities and locality of the Hamiltonians of the Geng-Xue equation (1.4)

In this section, we give two sequences of conserved quantities for the Geng-Xue equation (1.4), some of them seem to be new. Furthermore, we find all of them are homogeneous and all the Hamiltonians H_j, j ≤ −1 are local.

3.1. Conserved quantities

Defining

φ=(φ1,φ2,φ3)T, a=φ1φ2,b=φ3φ2,(3.1)

based on the Lax pair (1.5), we have

ρ=(lnφ2)x=λnb.(3.2)

The infinitely many conserved quantities for the Geng-Xue equation (1.4) can be constructed using the standard algorithm. Next we will utilize the procedure in Ref. [30] to obtain the required Hamiltonians.

Substituting these equalities (3.1) to the spectral problem (1.5) yields

ax=λm+b−aρ, bx=a−bρ,(3.3)

after expanding a, b into a series of the spectral parameter λ and equating the coefficients of different powers of λ in these equations (3.3), one can find a series of conserved densities from coefficients of ρ in power of λ.

Case 1: Expanding a, b as a = ∑_i_≥1 a_iλⁱ, b = ∑_j_≥1 b_jλ ^j and substituting them into the system (3.3), we establish that
a1=−ux,b1=−u,a2=0,b2=0,(1−∂2)b3=(nu2)x+nuux,a3=b3x+nb12,⋯,(1−∂2)ak=∑i,j≥1i+j=k−1[nbibj+(naibj)x],(1−∂2)bk=∑i,j≥1i+j=k−1[naibj+(nbibj)x].

Hence an infinite sequence of conserved quantities are obtained, and the first two are
H1=∫undx,H2=∫(uxv−uvx)undx.(3.4)

Remark 3.1.

The above two conserved quantities can be obtained from the Hamiltonian functionals (2.6) ℋ₁, ℋ₂ under the constraint N = 1, Q = m, R = n, U = u, V = v respectively.
Case 2: The second expansion is a=∑i≥0aiλ1−2i3, b=∑j≥0biλ−2j+13. Substituting it into the system (3.3) yields:
a0=m23n−13,b0=m13n−23,a1=−13(mn)−23mx,b1=13m−1n−2(mnx−mxn)⋯,ak+1=13[2(mn)−13(bk−1−akk)+bkk+∑i,j≥1i+j=k+1(nbibj−2b0−1aibj)],bk+1=13(mn)−23(bk−1−akx)−13(mn)−13[bkk+∑i,j≥1i+j=k+1(b0−1ai+nbi)bj].

In particular, we only consider the conserved quantities associated with b₃_k, k ∈ Z, and obtain the first two conserved quantities as
H−1=3∫(mn)13dx,H−2=127∫[5mx2nxn−53m−83−3(mn)−53mxxnx−18nxm−23n−53]dx.(3.5)

The negative powers of m and n that are involved in the above expressions for the conserved quantities raise an important structural property, whose solution established their analytic validity. Namely, we expect that if m > 0 and n > 0 hold initially (at time t = 0), then this property is preserved by the Geng-Xue flow. That this might be so is suggested by the granting, in the context of the CH equation, of such a property by means of a pointwise conservation law that has in that context very strong consequences for the long-time behavior of solutions [4, 20]. For the Geng-Xue flow we can proceed as follows. Define the smooth diffeomorphisms ψ(x,t) by ψ(x,0) = x and ψ_t(x,t) = u(ψ(x,t),t)v(ψ(x,t),t) for t > 0 and consider the expressions Q¯(x,t)=m(ψ(x,t),t)ψx3(x,t) and R¯(x,t)=n(ψ(x,t),t)ψx3(x,t). Since
ψxt=[ux(ψ,t)v(ψ,t)+u(ψ,t)vx(ψ,t)]ψx,t>0,
and ψ_x(x, 0) = 1 yield
ψx(x,t)=exp(∫0t[ux(ψ,s)v(ψ,s)+u(ψ,s)vx(ψ,s)]ds)>0,t≥0,
using (1.4) we can verify that the relations
∂tQ¯(x,t)=3u(ψ(x,t),t)vx(ψ(x,t),t)ψx3(x,t)Q¯(x,t),∂tR¯(x,t)=3ux(ψ(x,t),t)v(ψ(x,t),t)ψx3(x,t)R¯(x,t),(3.6)
hold for any fixed x. Therefore
Q¯(x,t)=Q¯(x,0)exp(3∫0t[u(ψ(x,s),s)vx(ψ(x,s),s)ψx3(x,s)ds),t≥0,R¯(x,t)=R¯(x,0)exp(3∫0t[ux(ψ(x,s),s)v(ψ(x,s),s)ψx3(x,s)ds),t≥0.(3.7)

These considerations prove that m(x, t) > 0 and n(x, t) > 0 at any t ≥ 0, provided that m(x, 0) > 0 and n(x, 0) > 0.

3.2. Homogeneous and local properties of the Hamiltonians

The first nontrivial negative flow in the Geng-Xue hierarchy is the Geng-Xue equation (1.4) [19]. It admits a bi-Hamiltonian structure [29], viz:

θt=KδH2δθ=JδH1δθ, θ=(mn),(3.8)

where H₁, H₂ are given by (3.4), and K, J are respectively the reductions of the Hamiltonian operators 𝒦, 𝒥 (2.4), (2.5) under the constraint N = 1, Q = m, R = n, U = u, V = v, namely

K=(0∂2−11−∂20),J=(J11J12J21J22),

with

J11=32m∂−1m+(mx+32m∂)(∂3−4∂)−1(mx+3m∂),J12=−32m∂−1n+(mx+32m∂)(∂3−4∂)−1(nx+3n∂),J21=−32n∂−1m+(nx+32n∂)(∂3−4∂)−1(mx+3m∂),J22=32n∂−1n+(nx+32n∂)(∂3−4∂)−1(nx+3n∂),

Furthermore, the first nontrivial positive flow in the hierarchy reads

mt=(m13n−23)xx−m13n−23,nt=n13m−23−(n13m−23)xx,

which can also be reformulated as a bi-Hamiltonian form

θt=JδH−2δθ=KδH−1δθ,(3.9)

herein H₋₁, H₋₂ are given by (3.5).

Hence, the Geng-Xue hierarchy may be constructed as

θtj=JδHjδθ=KδHj+1δθ,j∈Z,j≠0,−1,(3.10)

with infinitely many Hamiltonian functionals H_j generated by the recursive definition. It is immediately clear that all H_js are homogeneous and H_js, j ≥ 3 are nonlocal. Hence we will discuss the local property of H_j, j ≤ −1.

Lemma 3.1 ([26,34]).

If a differential function M^[θ] satisfies

∫M^[θ]dx=0

for all θ, then there exists a unique differential function N^[θ] up to addition of a constant such that M^[θ] is the total x-derivative M^[θ]=(N^[θ])x.

Theorem 3.1.

Let Xj[θ]=δHj[θ]δθ, then for each j ≤ −1, X_j[θ] and H_j[θ] are local.

Proof.

We only need to demonstrate the local property of X_js because of Hj[θ]=∫01Xj[ɛθ]⋅θdɛ. As j = −1, X₋₁[θ] is local since

X−1[θ]=(m−23n13,m13n−23)T.

Now suppose that X_j[θ] is local for j = −k. When j = −(k + 1), we have

X−(k+1)[θ]=J−1KX−k[θ]=(J−1K)kX−1[θ].(3.11)

To find an explicit expression for X₋_k[θ], a direct choice is to calculate the inverse operator of J. However, it seems to be difficult. Hence, we find an indirect way. Suppose h=(mn)13 and define

X−k[θ]=(Ak,Bk)T,Ek=(∂3−4∂)−1(3m∂+mx,3n∂+nx)X−k[θ],

it follows from the recursive relation (3.11) that

32m∂−1(mAk+1−nBk+1)+(32m∂+mx)Ek+1=(∂2−1)Bk,−32n∂−1(mAk+1−nBk+1)+(32n∂+nx)Ek+1=(1−∂2)Ak.

Eliminating E_k₊₁ from the above system and combating the definition of E_k₊₁ yield

(3m∂+mx3n∂+nxm−n)(Ak+1Bk+1)=13Θ(AkBk),(3.12)

where

Θ=(000∂2m(∂2−1))+(4∂−∂3∂2+∂2mx3m)1h∂−1(−m13n−23m−23n13)(1−∂2).

Then it is not difficult to solve A_k₊₁ and B_k₊₁ from the above system, and we find that A_k₊₁ and B_k₊₁ are local if there exist two differential functions M_k, N_k such that

Γ1=1h2[m(1−∂2)Ak+n(∂2−1)Bk]=Mkx,(3.13)

Γ2=1h([∂3−4∂+3(∂−2mx3m)∂(∂+2mx3m)]Mkh−(∂−2mx3m)∂6m(∂2−1)Bk)=Nkx.(3.14)

Using the Lemma 3.1, the equality (3.13) follows immediately from

∫Γ1dx=∫1h2[m(1−∂2)Ak+n(∂2−1)Bk]dx =∫X−1TKX−kdx =∫X−1TK(J−1K)k−1X−1dx =−∫X−1T(KJ−1)k−1KX−1dx =−∫X−1TK(J−1K)k−1X−1dx =−∫X−1TKX−kdx,

where we have used the skew-symmetric property of K, J. So ∫Γ₁dx = 0. Similarly, the equality (3.14) may be demonstrated from

∫Γ2dx=∫1h[∂3−4∂+(3∂−2mxm)(∂2+∂2mx3m)]Mkh −1h(3∂−2mxm)∂2m(∂2−1)Bkdx =∫f(1−∂2)Ak+g(∂2−1)Bkdx =∫(g,f)K(Ak,Bk)Tdx =−∫(Ak,Bk)K(g,f)Tdx =−∫J−1K(Ak,Bk)TJ(g,f)Tdx=0,

where

f=1h4(23mxx−43mnxxn−109mx2m+149mnx2n2+49mxnxn−2m),g=1h4(23nxx−43nmxxm−109nx2n+149nmx2m2+49mxnxn−2n).

In the above we have eliminated the derivatives of M_k in the integration by the equality (3.13) and used the skew-symmetric property of K, J, as well as the fact that (g, f)^T is a kernel of J. So all the Hamiltonians H_j, j ≤ −1 for the Geng-Xue equation (1.4) are local.

Acknowledgements

This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 11805071 and 11747010) and the Initial Founding of Scientific Research for the introduction of talents of Huaqiao University (Project No. 16BS513).

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TI  - Two-component generalizations of the Novikov equation
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Journal of Nonlinear Mathematical Physics

Two-component generalizations of the Novikov equation

1. Introduction

2. Two-component generalizations of the Novikov equation

2.1. The first two-component generalization of the Novikov equation

Remark 2.1.

2.2. The second two-component generalization of the Novikov equation

Remark 2.2.

3. Conserved quantities and locality of the Hamiltonians of the Geng-Xue equation (1.4)

3.1. Conserved quantities

Remark 3.1.

3.2. Homogeneous and local properties of the Hamiltonians

Lemma 3.1 ([26,34]).

Theorem 3.1.

Proof.

Acknowledgements

References

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