# Two Expanding Integrable Models of the Geng-Cao Hierarchy

Authors: Xiurong Guo; Yufeng Zhang; Xuping Zhang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

As far as linear integrable couplings are concerned, one has obtained some rich and
interesting results. In the paper, we will deduce two kinds of expanding integrable models of the Geng-Cao (GC) hierarchy by constructing different 6-dimensional Lie algebras. One expanding integrable model (actually, it is a nonlinear integrable coupling) reduces to a generalized Burgers equation and further reduces to the heat equation whose expanding nonlinear integrable model is generated. Another one is an expanding integrable model which is different from the first one. Finally, the Hamiltonian structures of the two expanding integrable models are obtained by employing the variational identity and the trace identity, respectively.


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## Body

## 1. Introduction

Integrable couplings are a kind of expanding integrable models of some known integrable hierarchies of equations. Based on this theory, one has obtained some integrable couplings of the known integrable hierarchies [1–8]. These integrable couplings are all linear with respect to the coupled variables. That is, if we introduce an evolution equation Ut=K(u), the coupled variable V satisfying Vt=S(u,v) is linear in V. The reason for this may be given by special Lie algebras. That is, such a Lie algebra G can be decomposed into a sum of the two subalgebras G1 and G2, which meets
(1)G=G1⊕G2,[G1,G2]⊂G2.If the subalgebraG2 is not simple, then the integrable coupling
(2)Ut=K(u),Vt=S(u,v)
is linear with respect to the variable V, which is obtained by introducing Lax pairs through the Lie algebra G. However, it is more interesting to seek for nonlinear integrable couplings because most of the coupled dynamics from physics, mechanics, and so forth are nonlinear. Recently, Ma and Zhu [9] introduced a kind of Lie algebra to deduce the nonlinear integrable couplings of the nonlinear Schrödinger equation and so forth, where the Lie subalgebras are simple and are different from the above. Based on this, Zhang [10] proposed a simple and efficient method for generating nonlinear integrable couplings and obtained the nonlinear integrable couplings of the Giachetti-Johnson (GJ) hierarchy and the Yang hierarchy, respectively. In addition, Zhang and Hon [11] proposed another Lie algebra which is different from those in [9, 10] to deduce nonlinear integrable couplings. Wei and Xia [12] also obtained some nonlinear integrable couplings of the known integrable hierarchies.In the paper, we want to start from a spectral problem proposed by Geng and Cao [13] to deduce an integrable hierarchy (called the GC hierarchy) under the frame of zero curvature equations by the Tu scheme [14] and obtain its new Hamiltonian structure. Then with the help of a 6-dimensional Lie algebra, a nonlinear expanding integrable model of the GC hierarchy is obtained, whose Hamiltonian structure is generated by the variational identity presented in [15]. The expanding integrable model can reduce to a generalized Burgers equation and further reduce to the heat equation. Another new 6-dimensional Lie algebra is constructed for which the second expanding integrable model is produced by using the Tu scheme whose Hamiltonian structure is derived from the trace identity proposed by Tu [14]. We shall find the two expanding integrable models of the GC hierarchy are different.


## 2. The GC Integrable Hierarchy and Its Hamiltonian Structure

We have known that(3)h=(100-1),e=(0100),f=(0010),
then one gets
(4)[h,e]=2e,[h,f]=-2f,[e,f]=h.
It is well known that span⁡{h,e,f}=A1 is a Lie algebra. A loop algebra of A1 is given by
(5)A1~=span⁡{h(n),e(n),f(n)},
where
(6)h(n)=h(0)λn,e(n)=e(0)λn,h(n)=h(0)λn,n∈Z.
By using the loop algebra A1~, introduce an isospectral problem [13]:
(7)U=(-λλuvλ)=-h(1)+ue(1)+vf(0).
Set
(8)V=V1h(0)+V2e(0)+V3f(0),
where
(9)Vi=∑m≥0Vimh(-m),i=1,2,3.
The stationary equation Vx=[U,V] admits that a solution for the V is as follows:
(10)(V1m)x=uV3m-vV2m,(V2m)x=-2V2,m+1-2uV1,m+1,(V3m)x=2V3,m+1+2vV1,m+1,
which gives rise to
(11)(V1,m+1)x=12u(V3m)x-12v(V2m)x.
Set
(12)V1,0=V2,0=V3,0=0,V1,1=α;
from (10) and (11) we have
(13)V2,1=-αu,V3,1=-αv,V1,2=-α2uv,V2,2=α2(ux+u2v),V3,2=α2(-vx+uv2),…
denote by
(14)V(n)=∑m=0n(V1mh(n-m)+V2me(n-m)+V3mf(n-m)).
We have
(15)-Vx(n)+[U,V(n)]=(2V2,n+1+2uV1,n+1)e(1)-Vx(n)+[U,V(n)]=-2(V3,n+1+2V1,n+1)f(0).
The compatibility of the following Lax pair
(16)U=-h(1)+ue(1)+vf(0)V(n)=∑m=0n‍(V1mh(n-m)+V2me(n-m)+V3mf(n-m))
gives rise to
(17)(uv)tn=(-2V2,n+1-2uV1,n+12V3,n+1+2vV1,n+1)=((V2n)x(V3n)x)=(0∂∂0)(V3nV2n)=J(V3nV2n),
where
(18)J=(0∂∂0)
is a Hamiltonian operator.By the trace identity presented in [14], we have
(19)〈V,∂U∂u〉=λV3,〈V,∂U∂v〉=λV2,〈V,∂U∂λ〉=-2λV1+uV3.
Substituting the above results to the trace identity yields
(20)δδw(-2λV1+uV3)=λ-γ∂∂λλγ(λV3λV2),
where
(21)δδw=(δδu,δδv)T.
Comparing the coefficients of λ-n of both sides in (20) gives
(22)δδw(-2V1,n+1+uV3,n+1)=(-n+1+γ)(λV3nλV2n).
It is easy to see γ=-1. Thus, we have
(23)(λV3nλV2n)=δδu(2V1,n+1-uV3,n+1n)=δHnδu,
where
(24)Hn=2V1,n+1-uV3,n+1n
are Hamiltonian conserved densities of the Lax integrable hierarchy (17). Therefore, we get a Hamiltonian form of the hierarchy (17) as follows:
(25)(uv)tn=JδHnδu.
Let us consider the reduced cases of (17). When n=1, we get that
(26)ut1=-αux,vt1=-αvx.
Taking n=2, one gets a generalized Burgers equation:
(27)ut2=α2uxx+αuuxv+α2u2vx,vt2=-α2vxx+α2uxv2+αuvvx.Remark 1.
The Hamiltonian structure (23) is different from that in [14]. We call (17) the GC hierarchy.


## 3. The First Expanding Integrable Model of the GC Hierarchy

Zhang and Tam [16] proposed a few kinds of Lie algebras to deduce nonlinear integrable couplings. In the section we will choose one of them to investigate the nonlinear integrable coupling of the hierarchy (17).Consider the following Lie algebra:(28)F=span⁡{f1,…,f6},
where
(29)f1=(e100e1),f2=(e200e2),f3=(e300e3),f4=(0e10e1),f5=(0e20e2),f6=(0e30e3),e1=(100-1),e2=(0100),e3=(0010).
Define
(30)[a,b]=ab-ba,∀a,b∈F.
It is easy to compute that
(31)[f1,f2]=2f2,[f1,f3]=-2f3,[f2,f3]=f1,[f1,f4]=0,[f1,f5]=2f5,[f1,f6]=-2f6,[f2,f4]=-2f5,[f2,f5]=0,[f2,f6]=f4,[f3,f4]=2f6,[f3,f5]=-f4,[f3,f6]=0,[f4,f5]=2f5,[f4,f6]=-2f6,[f5,f6]=f4.
Set
(32)F1=span⁡{f1,f2,f3},F2=span⁡{f4,f5,f6};
we have
(33)F1=F1⊕F2,[F1,F2]⊂F2;F1 and F2 are all simple Lie-subalgebras of the Lie algebra F. The corresponding symmetric constant matrix M appearing in the variational identity is that
(34)(2η1002η20000η100η20η100η202η2002η20000η200η20η200η20).A loop algebra corresponding to the Lie algebraF is defined by
(35)F~=span⁡{f1(n),…,f6(n)},fi(n)=fiλn,[fi(m),fj(n)]=[fi,fj]λm+n,1≤i,j≤6,m,n∈Z.
We use the loop algebra F~ to introduce a Lax pair:
(36)U=-f(1)+uf2(1)+vf3(0)+u1f5(1)+u2f6(0),V=∑m≥0(V1mf1(1-m)+V2mf2(1-m)+V3mf3(-m)BBBBB+V4mf4(1-m)+V5mf5(1-m)+V6mf6(-m)).The stationary equationVx=[U,V] is equivalent to
(37)(V1m)x=uV3m-vV2m,(V2m)x=-2V2,m+1-2uV1,m+1,(V3m)x=2V3,m+1+2vV1,m+1,(V4m)x=u1V3m-u2V2m-(v+u2)V5m+(u+u1)V6m,(V5m)x=-2V5,m+1-2u1V1,m+1-2(u+u1)V4,m+1,(V6m)x=2V6,m+1+2u2V1,m+1+2(v+u2)V4,m+1
from which we have
(38)(V1,m+1)x=12u(V3m)x-12v(V2m)x,(V4,m+1)=12u1(V3m)x+12u2(V2m)x(V4,m+1)=+12(u+u2)(V5m)x+12(u+u1)(V6m)x.
Set
(39)V1,0=V2,0=V3,0=V4,0=V5,0=V6,0=0,V1,1=α;
we obtain from (37)
(40)V3,1=-αv,V2,1=-αu,V4,1=0,V4,1=-αu1,V6,1=-αu2,V1,2=-α2uv,V4,2=-α2(u1v+uv2+u1u2),V5,2=α2(u1x+u1uv+(u+u1)(u1v+uu2+u1u2)),V6,2=α2(-u2x+u2uv+(v+u2)(u1v+uu2+u1u2)),…Note(41)V(n)=∑m=0n(V1mf1(1+n-m)+V2mf2(1+n-m)bbbbbb+V3mf3(-m)+V4mf4(1+n-m)bbbbbb+V5mf5(1+n-m)+V6mf6(n-m));
a direct calculation yields
(42)-Vx(n)+[U,V(n)]=-2(V3,n+1+vV1,n+1)f3(0)BBBBB+(2V2,n+1+2uV1,n+1)f2(1)BBBBB-2((v+u2)V4,n+1+u2)V1,n+1+V6,n+1f6(0)BBBBB+2((u+u1)V4,n+1+u1)V1,n+1+V5,n+1f5(1)=-(V3n)xf3(0)-(V2n)xf2(1)BBBB-(V6n)xf6(0)-(V5n)xf5(1).
Therefore, zero curvature equation
(43)Ut-Vxn+[U,V(n)]=0
admits that
(44)(uvu1u2)=((V2n)x(V3n)x(V5n)x(V6n)x).
Set u1=u2=0, (44) reduces to the integrable hierarchy (17). When we take n=2, we get an expanding nonlinear integrable model of the generalized Burgers equation (27) as follows:
(45)ut2=α2uxx+αuuxv+α2u2vx,vt2=-α2vxx+α2uxv2+αuvvx,u1t2=α2(u1xx+u1xuv+u1(uv)xBBBbbBbbbb+(u+u1)x(u1v+uu2+u1u2)BBBibBbbbb+(u+u1)(u1v+uu2+u1u2)x),u2t2=α2(-u2xx+u2xuv+u2(uv)xbbbbbbbbbbi+(v+u2)x(u1v+uu2+u1u2)bbbbbbibbbi+(v+u2)(u1v+uu2+u1u2)x).
Obviously, the coupled equations are nonlinear with respect to the coupled variables u1 and u2. Therefore, the hierarchy (44) is a nonlinear expanding integrable model of the integrable system (17); actually, it is a nonlinear integrable coupling.The nonlinear expanding integrable model (45) can be written as two parts, one is just right (27); another one is the latter two equations in (45), which can be regarded as a coupled nonlinear equation with variable coefficients u, v, and their derivatives in the variable x, where the functions u, v satisfy (27). In particular, we take a trivial solution of (27) to be u=v=0; then (45) reduces to the following equations:
(46)u1t2=α2[u1,xx+(u12u2)x],u2t2=α2[-u2,xx+(u22u1)x].
When we set u2=0, the above equations reduce to the well-known heat equation.In order to deduce Hamiltonian structure of the nonlinear integrable coupling (44), we define a linear functional [11]:
(47){a,b}=aTMb,
where a=(a1,…,a6)T, b=(b1,…,b6)T.It is easy to see that the Lie algebraF is isomorphic to the Lie algebra R6 if equipped with a commutator as follows:
(48)[a,b]T=(a2b3-a3b2,2a1b2-2a2b1,2a3b1-2a1b3,a2b6-a6b2+a5b3-a3b5+a5b6-a6b5,2a1b5-2a5b1+2a4b2-2a2b4+2a4b5-2a5b4,2a3b4-2a4b3+2a6b1-2a1b6+2a6b4-2a4b6).
Thus, under the Lie algebra R6, the Lax pair (36) can be written as
(49)U=(-λ,uλ,v,0,u1,λ,u2)T,V=(V1λ,V2λ,V3,V4λ,V5λ,V6)T.
In terms of (48) and (49) we obtain that
(50){V,∂U∂u}=(η1V3+η2V6)λ,{V,∂U∂v}=(η1V2+η2V5)λ,{V,∂U∂u1}=(η2V3+η2V6)λ,{V,∂U∂u2}=(η2V2+η2V5)λ,{V,∂U∂λ}=-2η1V1+(η1u+η2u1)V3-2η2V4+(η2u+η2u1)V6.
Substituting the above results into the variational identity yields
(51)δδw∫x(-2η1V1+(η1u+η2u1)V3+η2(u+u1)V6)dx=λ-γ∂∂λλγ((η1V3+η2V6)λ(η1V2+η2V5)λ(η2V3+η2V6)λ(η2V2+η2V5)λ),
where
(52)δδw=(δδu,δδv,δδu1,δδu2)T.
Comparing the coefficients of λ-n on both sides in (51) gives
(53)δδw∫x(-2η1V1,n+1+(η1u+η2u1)V3,n+1bbbbbbb+2η2V4,n+1+η2(u+u1)V6,n+1)dx=(-n+1+γ)(η1V3n+η2V6nη1V2n+η2V5nη2V3n+η2V6nη2V2n+η2V5n).
From (37) we have γ=-1. Thus, we get that
(54)(η1V3n+η2V6nη1V2n+η2V5nη2V3n+η2V6nη2V2n+η2V5n)=δHn+1δu,
where
(55)Hn+1=∫1/nx‍(2η1V1,n+1-(η1u+η2u1)V3,n+1bbbbbbb+2η2V4,n+1-η2(u+u1)V6,n+1)dx.
Therefore, we obtain the Hamiltonian structure of the nonlinear integrable coupling (44) as follows:
(56)δδw∫x(-2η1V1,n+1+(η1u+η2u1)V3,n+1bbbbbbbb-2η2V4,n+1+η2(u+u1)V6,n+1)dx,Wtn=(uvu1u2)tn=(0-∂η1-η20∂η1-η2-∂η1-η20∂η1-η200∂η1-η20η1∂(η1-η2)η2∂η1-η20η1∂(η1-η2)η20)×(η1V3n+η2V6nη1V2n+η2V5nη2V3n+η2V6nη2V2n+η2V5n)=J(η1V3n+η2V6nη1V2n+η2V5nη2V3n+η2V6nη2V2n+η2V5n)=JδHn+1δu,
where J is obviously Hamiltonian.


## 4. The Second Expanding Integrable Model of the GC Hierarchy

In this section we construct a new 6-dimensional Lie algebra to discuss the second integrable coupling of the GC hierarchy. Set(57)h1=f1,h2=f2,h3=f3,hj=(0ej-3ej-30),j=4,5,6.
It is easy to see that
(58)[h1,h2]=2h2,[h1,h3]=-2h3,[h2,h3]=h1,[h1,h4]=0,[h1,h5]=2h5,[h1,h6]=-2h6,[h2,h4]=-2h5,[h2,h5]=0,[h2,h6]=h4,[h3,h4]=2h6,[h3,h5]=-h4,[h3,h6]=0,[h4,h5]=2h2,[h4,h6]=-2h3,[h5,h6]=h1.
If we set G=span⁡{h1,…,h6}, G1=span⁡{h1,h2,h3}, and G2=span⁡{h4,h5,h6}, then we have that
(59)G=G1+G2,[G1,G2]notinG2.
Hence, the integrable couplings of the GC hierarchy cannot be generated by the Lie algebra G as above under the frame of the Tu scheme. In what follows, we will deduce a nonlinear expanding integrable model of the GC hierarchy.Set(60)U=-h1(1)+uh2(1)+vh3(0)+w1h5(1)+w2h6(0),V=∑m≥0(V1mh1(1-m)+V2mh2(1-m)+V3mh3(-m)bbbbbbb+V4mh4(1-m)+V5mh5(1-m)bbbbbbb+V6mh6(1-m)),
where hi(m)=hiλm, i=1,2,3,4,5,6.Solving the stationary zero curvature equation(61)Vx=[U,V]
gives rise to
(62)(V1m)x=uV3m-vV2m-w2V5m+w1V6m,(V2m)x=-2V2,m+1-2uV1,m+1-2w1V4,m+1,(V3m)x=2V3,m+1+2vV1,m+1+2w2V4,m+1,(V4m)x=uV6m-vV5m+w1V3m-w2V2m,(V5m)x=-2V5,m+1-2uV4,m+1-2w1V1,m+1,(V6m)x=2V6,m+1+2vV4,m+1+2w2V1,m+1.
Let V2,1=-αu, V3,1=-αv, V5,1=-αw1, and V6,1=αw2; then one gets from (62) that
(63)V1,2=-α2uv,V2,2=α2(ux+u2v),V3,2=α2(-vx+uv2),V4,1=0,V4,2=α2(uw2-w1v),V5,2=α2(w1,x-u2w2+w1uv),V6,2=α2(w2,x+v2w1),….
Noting V+(n)=∑m=0n(V1mh1(1+n-m)+V2mh2(1+n-m)+V3mh3(n-m)+V4mh4(1+n-m)+V5mh5(1+n-m)+V6mh6(n-m))=λnV-V-(n), one infers that -V+,x(n)+[U,V+(n)]=(2V2,n+1+2uV1,n+1+2w1V4,n+1)h2(1)-2(V3,n+1+vV1,n+1+w2V4,n+1)h3(0)+(2V5,n+1+2uV4,n+1+2w1V1,n+1)h5(1)-2(V6,n+1+vV4,n+1+w2V1,n+1)h6(0).SetV(n)=V+(n), by employing the zero curvature equation
(64)Utn-Vx(n)+[U,V(n)]=0
we have
(65)u-tn=(uvw1w2)tn=(-2V2,n+1-2uV1,n+1-2w1V4,n+12V3,n+1+2vV1,n+1+2w2V4,n+1-2V5,n+1-2uV4,n+1-2w1V1,n+12V6,n+1+2vV4,n+1+2w2V1,n+1).
When n=2, α=2, (65) reduces to
(66)ut2=uxx+(u2v)x,vt2=-vxx+(uv2)x,w1,t2=w1,xx-(u2w2)x+(w1uv)x,w2,t2=w2,xx+(v2w1)x.
It is remarkable that (66) is linear with respect to the variables w1, w2; however, it is nonlinear.Equation (60) can be written as
(67)U=(-λuλ0w1λvλw200w1λ-λuλw20vλ),V=(V1λV2λV4λV5λV3-V1λV6-V4λV4λV5λV1λV2λV6-V4λV3-V1λ).
By computing that
(68)∂U∂u=(0λ000000000λ0000),∂U∂v=(0000100000000010),∂U∂w1=(000λ00000λ000000),∂U∂w2=(0000001000001000),∂U∂λ=(-1u0w101000w1-1u0001),
thus, we have
(69)〈V,∂U∂u〉=2λV3,〈V,∂U∂v〉=2λV2,〈V,∂U∂w1〉=2λV6,〈V,∂U∂w2〉=2λV5,〈V,∂U∂λ〉=-4λV1+2uV3+2w1V6.Substituting the above consequences into the trace identity proposed by Tu [14] yields that
(70)δδu-(-4λV1+2uV3+2w1V6)=λ-γ∂∂λλγ(2λV32λV22λV62λV5).
Comparing the coefficients of λ-n gives
(71)δδu-(-4V1,n+2uV3,n-1+2w1V6,n-1)=(-n+1+γ)(2V3n2V2n2V6n2V5n).Inserting the initial values in (62) gives γ=-1. Therefore, we obtain that
(72)(2V3n2V2n2V6n2V5n)=δδu-(4V1n-2uV3,n-1-2w1V6,n-1n)≡δHnδu-,
where Hn=(1/n)(4V1n-2uV3,n-1-2w1V6,n-1) are conserved densities of the expanding integrable model (65). Thus, (65) can be written as the Hamiltonian structure
(73)u-tn=(uvw1w2)tn=JδHnδu-,
where J=(1/2)(0∂00∂000000∂00∂0), ∂=∂/∂x, is a Hamiltonian operator.



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*Source: 860935-2014-02-27.xml*