New Applications of a Kind of Infinitesimal-Operator Lie Algebra

Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2016, Article ID 7639013, 12 pages http://dx.doi.org/10.1155/2016/7639013

Research Article New Applications of a Kind of Infinitesimal-Operator Lie Algebra Honwah Tam,1 Yufeng Zhang,2 and Xiangzhi Zhang2 1

Department of Computer Science, Hong Kong Baptist University, Kowloon Tong, Hong Kong College of Sciences, China University of Mining and Technology, Xuzhou 221116, China

2

Correspondence should be addressed to Yufeng Zhang; [email protected] Received 22 January 2016; Accepted 26 June 2016 Academic Editor: Remi L´eandre Copyright © 2016 Honwah Tam et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Applying some reduced Lie algebras of Lie symmetry operators of a Lie transformation group, we obtain an invariant of a second-order differential equation which can be generated by a Euler-Lagrange formulism. A corresponding discrete equation approximating it is given as well. Finally, we make use of the Lie algebras to generate some new integrable systems including (1 + 1) and (2 + 1) dimensions.

1. Introduction Some research on difference equations admitting Lie-point transformations can be found in the literature [1, 2]. Specifically, some symmetry-preserving difference schemes for nonlinear differential equations were discovered in [3–5]. We first briefly recall some fundamental notations. Given 𝐿 satisfying the Euler-Lagrange equation, 𝜕𝐿 𝛿𝐿 𝜕𝐿 = − 𝐷 ( 󸀠 ) = 0, 𝛿𝑦 𝜕𝑦 𝜕𝑦

(1)

𝜕 𝜕 𝜕 + 𝑦󸀠 + 𝑦󸀠󸀠 󸀠 + ⋅ ⋅ ⋅ 𝜕𝑥 𝜕𝑦 𝜕𝑦

̃ = ∫ 𝐿 (𝑥, 𝑦, 𝑦 ) 𝑑𝑥, 𝐿 Ω

1

Ω⊂𝑅

(3)

𝑦󸀠󸀠 = 𝑓 (𝑥, 𝑦, 𝑦󸀠 ) .

(4)

Group 𝐺 generated by the vector field 𝑋 = 𝜉 (𝑥, 𝑦)

𝜕 𝜕 + 𝜂 (𝑥, 𝑦) 𝜕𝑥 𝜕𝑦

pr 𝑋 (𝐿) + 𝐿𝐷 (𝜉) = 𝐷 (𝑉) ,

(5)

(7)

then the group 𝐺 is known as an infinitesimal divergence symmetry. By Noether’s theorem, one infers that [5] 𝛿𝐿 𝛿𝑦

𝜕𝐿 + 𝐷 (𝜉𝐿 + (𝜂 − 𝜉𝑦 ) 󸀠 ) . 𝜕𝑦

(8)

󸀠

Therefore, if the vector operator 𝑋 is a divergence symmetry, we have a first integral 𝐾 = 𝜉𝐿 + (𝜂 − 𝜉𝑦󸀠 )

achieves its extremal values on [1]. Equation (1) is an ODE that can be rewritten as

(6)

If there exists a function 𝑉(𝑥, 𝑦) such that

(2)

is a Lie group operator, it is well-known that the functional 󸀠

pr 𝑋 (𝐿) + 𝐿𝐷 (𝜉) = 0.

pr 𝑋 (𝐿) + 𝐿𝐷 (𝜉) = (𝜂 − 𝜉𝑦󸀠 )

where 𝐷=

̃ if and only if the is a variational symmetry of the functional 𝐿 Lagrangian satisfies

𝜕𝐿 − 𝑉, 𝜕𝑦󸀠

(9)

which corresponds to the Euler-Lagrange equation. In [4], the Lagrangian formalism for second-order difference equations was reviewed. Consider a finite difference functional ̂ = ∑L (𝑥, 𝑥+ , 𝑦, 𝑦+ ) ℎ+ 𝐿 (10) Ω

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defined on some one-dimensional lattice Ω with step spacing ℎ+ , which can be represented by ℎ+ = 𝜑 (𝑥, 𝑦, 𝑥+ , 𝑦+ ) .

(11)

For an arbitrary curve the stationary value of a differential functional is given by any solution of the quasiextremal equations [4, 5]: 𝜕L 𝜕L− 𝛿L = ℎ+ + ℎ− = 0, 𝛿𝑦 𝜕𝑦 𝜕𝑦 𝛿L 𝜕L 𝜕L− = ℎ+ + ℎ− + L− − L. 𝛿𝑥 𝜕𝑥 𝜕𝑥

(12)

+ℎ

+ℎ

(13)

then each element 𝑋 of the Lie algebra corresponding to 𝐺 provides us with a first integral of (12): 𝐾 = ℎ− 𝜂

𝜕L− 𝜕L− + ℎ− 𝜉 + 𝜉L− − 𝑉. 𝜕𝑦 𝜕𝑥

2. A Few Lie Subalgebras of the Operator Algebra The Lie algebra (15) has the following commutative operations [6]: [𝑋1 , 𝑋2 ] = 0,

It can be verified that if the Lagrangian density L is divergence invariant under group 𝐺, it holds that pr 𝑋 (L) + L𝐷 (𝜉) = 𝐷 (𝑉) ;

under the framework of the Tu scheme. Finally, we establish a vector Lie algebra to generate expanding integrable models of the (1 + 1)-dimensional and (2 + 1)-dimensional integrable hierarchies of evolution equations obtained in the paper.

(14)

[𝑋1 , 𝑋3 ] = −𝑋1 , [𝑋1 , 𝑋4 ] = −𝑋2 , [𝑋1 , 𝑋5 ] = −2𝑋3 − 𝑋7 , [𝑋1 , 𝑋6 ] = [𝑋1 , 𝑋7 ] = 0, [𝑋1 , 𝑋8 ] = −𝑋6 , [𝑋2 , 𝑋3 ] = [𝑋2 , 𝑋4 ] = 0,

In the paper, we start from the infinitesimal symmetry operators of an eight-parameter projective transformation group in 𝑅2 [6],

[𝑋2 , 𝑋5 ] = −𝑋4 ,

𝜕 𝜕 𝑋1 = 𝑥 + 𝑥𝑦 , 𝜕𝑥 𝜕𝑦

[𝑋2 , 𝑋7 ] = −𝑋2 ,

2

𝑋2 = 𝑥𝑦

[𝑋2 , 𝑋8 ] = −𝑋3 − 2𝑋7 ,

𝜕 𝜕 + 𝑦2 , 𝜕𝑥 𝜕𝑦

[𝑋3 , 𝑋4 ] = −𝑋4 ,

𝜕 𝑋3 = 𝑥 , 𝜕𝑥 𝑋4 = 𝑦 𝑋5 =

𝜕 , 𝜕𝑥

𝜕 , 𝜕𝑥

𝑋6 = 𝑥

𝜕 , 𝜕𝑦

𝑋7 = 𝑦

𝜕 , 𝜕𝑦

𝑋8 =

[𝑋2 , 𝑋6 ] = −𝑋1 ,

(16)

[𝑋3 , 𝑋5 ] = −𝑋5 , [𝑋3 , 𝑋6 ] = 𝑋6 , (15)

𝜕 , 𝜕𝑦

to discuss some continuous symmetries of Lagrangians and general solutions of discrete equations. We generate new (1 + 1)-dimensional and (2 + 1)-dimensional integrable systems with three potential functions. Specifically, we obtain an integrable coupling of the standard Burgers equation and a (2 + 1)-dimensional integrable coupling of the heat conduct equation. We also derive a (2 + 1)-dimensional integrable coupling of the (2 + 1)-dimensional hyperbolic equation

[𝑋3 , 𝑋7 ] = [𝑋3 , 𝑋8 ] = [𝑋4 , 𝑋5 ] = 0, [𝑋4 , 𝑋6 ] = 𝑋7 − 𝑋3 , [𝑋4 , 𝑋7 ] = −𝑋4 , [𝑋4 , 𝑋8 ] = −𝑋5 , [𝑋5 , 𝑋6 ] = 𝑋8 , [𝑋5 , 𝑋7 ] = [𝑋5 , 𝑋8 ] = 0, [𝑋6 , 𝑋7 ] = 𝑋6 , [𝑋6 , 𝑋8 ] = 0, [𝑋7 , 𝑋8 ] = −𝑋8 . From this Lie algebra, some interesting Lie algebras given in [4] can be obtained. In fact, the Lie algebras in [4] denoted by

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𝐷2,1 , 𝐷2,2 , 𝐷3,1 , 𝐷3,2 , 𝐷3,3 are actually linear combinations of the elements in (15):

is a Lie subalgebra of 𝑔, with the commutative relations [𝑋5 , 𝑋6 ] = 𝑋8 ,

𝐷2,1 = {𝑋5 , 𝑋8 } ,

[𝑋5 , 𝑋7 ] = [𝑋5 , 𝑋8 ] = 0,

𝐷2,2 = {𝑋3 + 𝑋7 , 𝑋8 } , 𝐷3,1 = {𝑋5 , 𝑋8 , 𝑋3 + 𝑘𝑋7 } ,

[𝑋5 , 𝑋] = 𝑋5 + 𝑘𝑋8 , (17)

[𝑋6 , 𝑋8 ] = 0,

𝐷3,2 = {𝑋5 , 2𝑋3 + 𝑋7 , 𝑋1 } ,

[𝑋6 , 𝑋7 ] = 𝑋6 ,

𝐷3,3 = {𝑋5 , 𝑋8 , 𝑋3 + 𝑋6 + 𝑋7 } .

[𝑋6 , 𝑋] = −𝑋8 ,

In [4], by using some subalgebras, some difference equations corresponding to Lagrangian invariants and some general solutions of the corresponding quasiextremal equations were obtained. Therefore, it would be important to further study the operator Lie algebra (15) for applications in generating new difference equations and new integrable systems. If we denote the Lie algebra (15) by 𝑔 = span {𝑋1 , 𝑋2 , . . . , 𝑋8 } ,

(23)

[𝑋7 , 𝑋] = −𝑘𝑋6 , [𝑋7 , 𝑋8 ] = −𝑋8 , [𝑋8 , 𝑋] = 𝑋7 . If we let 𝑘 = −1 and remove the element 𝑋5 from the Lie algebra 𝑔5 , we get a 4-dimensional Lie subalgebra

(18)

then it is easy to have

𝑔4 = span {𝑋6 , 𝑋7 , 𝑋8 , 𝑋} .

(24)

The corresponding commutative operations are given by

𝑔 = 𝑔1 + 𝑔2 , 𝑔1 = span {𝑋1 , 𝑋2 , 𝑋3 , 𝑋4 } ,

[𝑋6 , 𝑋7 ] = 𝑋6 , [𝑋6 , 𝑋8 ] = 0,

(19)

𝑔2 = span {𝑋5 , 𝑋6 , 𝑋7 , 𝑋8 } ,

[𝑋6 , 𝑋] = −𝑋8 ,

where 𝑔1 and 𝑔2 are Lie subalgebras of 𝑔 and both are not semisimple. In addition, we have

[𝑋7 , 𝑋8 ] = −𝑋8 ,

(25)

[𝑋7 , 𝑋] = 𝑋6 ,

[𝑔1 , 𝑔1 ] ⊂ 𝑔1 ,

[𝑋8 , 𝑋] = 𝑋7 .

[𝑔2 , 𝑔2 ] ⊂ 𝑔2 ,

(20) [𝑔1 , 𝑔2 ] not in 𝑔1 , 𝑔2 .

Hence, the Lie algebra 𝑔 is not a direct sum of the Lie subalgebras 𝑔1 and 𝑔2 . Because of relation (20), 𝑔 is not a symmetric Lie algebra. Therefore, 𝑔 itself cannot be utilized to generate integrable couplings. However, subalgebras of 𝑔 have the potential and will be investigated in the following. In addition, some new Lie subalgebras can be obtained from (15). For example, we take 𝑋3 ∈ 𝑔1 and 𝑋5 , 𝑋6 , 𝑋7 ∈ 𝑔2 to make a linear combination: 𝑋 = 𝑋3 + 𝑋5 + 𝑘𝑋6 + 𝑋7 = (1 + 𝑥)

𝜕 𝜕 + (𝑘𝑥 + 𝑦) . 𝜕𝑥 𝜕𝑦

(21)

3. General Solutions of Some Difference Equations A direct calculation yields a second-order ODE corresponding to the vector field 𝑋 on the space (𝑥, 𝑦, 𝑦󸀠 , 𝑦󸀠󸀠 ): 𝑦󸀠󸀠 =

(1−2𝑘)/(1−𝑘) 1 [𝑘 + (1 − 𝑘) 𝑦󸀠 ] . 1 − 2𝑘

(26)

This equation can be obtained from the Lagrangian

It is easy to verify that 𝑔5 = span {𝑋5 , 𝑋6 , 𝑋7 , 𝑋8 , 𝑋}

In what follows, we shall make use of the Lie subalgebras (22) and (24) to generate second-order differential equations and difference equations that preserve Lie-point symmetries. We derive the general solutions of the corresponding quasiextremal equations. Furthermore, we employ the Lie subalgebras to generate some new (1 + 1)-dimensional and (2 + 1)-dimensional integrable hierarchies of evolution equations.

(22)

𝐿=𝑦+

1/(1−𝑘) 1 − 2𝑘 . [𝑘 + (1 − 𝑘) 𝑦󸀠 ] 𝑘

(27)

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Obviously, the Lagrangian admits the symmetries 𝑋5 and 𝑋8 : pr 𝑋5 (𝐿) + 𝐿𝐷 (𝜉5 ) = 0, pr 𝑋8 (𝐿) + 𝐿𝐷 (𝜉8 ) = 1 = 𝐷 (𝑥) .

(28)

With the help of Noether’s theorem we obtain the following first integrals: 𝐽1 = 𝑦 +

1/(1−𝑘) 1 − 2𝑘 [𝑘 + (1 − 𝑘) 𝑦󸀠 ] 𝑘

𝑘/(1−𝑘) 1 − 2𝑘 󸀠 − ≡ 𝐴 0, 𝑦 [𝑘 + (1 − 𝑘) 𝑦󸀠 ] 𝑘

𝐽2 =

(29)

𝐷 (𝑦) ≡ 𝑦𝑥 = −

−ℎ

𝐼1 =

+

𝑘 2𝑘 − 1 [ (𝑥 + 𝐵0 )] 𝑘 (1 − 𝑘) 1 − 𝑘 ⋅ [(

(30)

which is the general solution to (26). Let us take the difference Lagrangians (31)

which satisfies pr 𝑋5 (L) + L𝐷 (𝜉5 ) = 0,

(32)

pr 𝑋8 (L) + L𝐷 (𝜉8 ) = 𝐷 (𝑥) ,

(33)

+ℎ

1 − 2𝑘 𝑘/(1−𝑘) ≡ 𝐵, [𝑘 + (1 − 𝑘) 𝑦𝑥 ] 𝑘

𝑥+2 1 − 2𝑘 𝑘/(1−𝑘) + 𝑥 + [𝑘 + (1 − 𝑘) 𝑦𝑥 ] 2 𝑘 −

(1 − 2𝑘)2 [𝑘 + (1 − 𝑘)] 𝑦𝑥 ≡ 𝐶. 𝑘 (1 − 𝑘)

(1−𝑘)/𝑘 1 𝑘 (1 − 2𝑘)2 𝐵𝑥+ − 𝑥+2 − [ (𝐵 − 𝑥+ )] 2 𝑘 (1 − 𝑘) 1 − 2𝑘

Inserting 𝑦𝑥 into the first equation in (37) gives rise to the general solution to the quasiextremal equation (35): 𝑦+ = 𝑦 +

𝐷 (𝑦) ≡ 𝑦𝑥 =

𝑦 − 𝑦 𝑦+ − 𝑦 1 . = ( 𝑆 − 1) (𝑦) = + ℎ+ +ℎ 𝑥+ − 𝑥 ℎ+

Then the variations of L yield the quasiextremal equations

+

2𝑘 − 1 𝑘/(1−𝑘) + 𝑦𝑥 [𝑘 + (1 − 𝑘) 𝑦𝑥 ] 𝑘 +

1 − 2𝑘 1/(1−𝑘) [𝑘 + (1 − 𝑘) 𝑦𝑥 ] 𝑘

+

2𝑘 − 1 1/(1−𝑘) = 0, [𝑘 + (1 − 𝑘) 𝑦𝑥 ] 𝑘

(1 − 𝑘) (1 − 2𝑘)𝑘−1

[𝑘 (39)

which is defined on the lattice determined by the lattice equation (38). We now consider the invariant model for (26). It is easy to see that (26) admits a Lie transformation group that can be represented by the Lie algebra 𝑔4 . The group consists of infinitesimal vector fields. The corresponding prolongation operators are given by

1 − 2𝑘 𝑘/(1−𝑘) = 0, [𝑘 + (1 − 𝑘) 𝑦𝑥 ] 𝑘

𝛿L 𝑦+ − 𝑦− 1 − 2𝑘 𝑘/(1−𝑘) : + 𝑦𝑥 [𝑘 + (1 − 𝑘) 𝑦𝑥 ] 𝛿𝑥 2 𝑘

2𝑘𝑘−1 𝐴𝑘

(1−𝑘)/𝑘 𝑘 −( ], (𝐵 − 𝑥+ )) 1 − 2𝑘

(34)

𝛿L ℎ+ + ℎ− 2𝑘 − 1 𝑘/(1−𝑘) : + [𝑘 + (1 − 𝑘) 𝑦𝑥 ] 𝛿𝑦 2 𝑘

(38)

≡ 𝐶.

where +ℎ

(37)

Solving 𝑦𝑥 from the second equation in (37) and then substituting the solution into the third one yields the lattice equation:

(1−𝑘)/𝑘 𝑘 − 𝑘] , (𝑥 + 𝐵0 )) 1 − 2𝑘

1/(1−𝑘) 𝑦 + 𝑦+ 1 − 2𝑘 , + [𝑘 + (1 − 𝑘) 𝐷 (𝑦)] +ℎ 2 𝑘

1 − 2𝑘 1/(1−𝑘) ≡ 𝐴, [𝑘 + (1 − 𝑘) 𝑦𝑥 ] 𝑘

𝐼2 = 𝑥+ + 𝐼3 =

((1−𝑘)/𝑘)2

(36)

𝑦 + 𝑦+ 2𝑘 − 1 𝑘/(1−𝑘) + 𝑦𝑥 [𝑘 + (1 − 𝑘) 𝑦𝑥 ] 𝑘 2 +

1/𝑘 2𝑘 − 1 𝑘 𝑦 (𝑥) = 𝐴 0 + [ (𝑥 + 𝐵0 )] 𝑘 1 − 2𝑘

𝑦 − 𝑦− 𝑦 − 𝑦− 1 = ( 𝑆 − 1) (𝑦) = ℎ− −ℎ 𝑥 − 𝑥− ℎ−

and ℎ+ , ℎ− are spatial spacing steps. The case ℎ+ = ℎ− implies that the grid mesh is uniform. By using formula (14), the first integrals are obtained:

𝑘/(1−𝑘) 1 − 2𝑘 − 𝑥 ≡ 𝐵0 . [𝑘 + (1 − 𝑘) 𝑦󸀠 ] 𝑘

Solving the second equation for 𝑦󸀠 in (29) and substituting it into the first equation, we obtain

L=

where

(35) ̂ 6 = pr 𝑋6 𝑋 =𝑥

𝜕 𝜕 𝜕 + 𝑥+ + + ℎ+ 𝐷 (𝑥) +ℎ 𝜕𝑦 𝜕𝑦 𝜕ℎ+

+ ℎ− 𝐷 (𝑥) −ℎ

𝜕 , 𝜕ℎ−

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̂ 7 = 𝑦 𝜕 + 𝑦+ 𝜕 + [𝐷 (𝑦) − 𝑦1 𝐷 (0)] 𝜕 𝑋 +ℎ 𝜕𝑦 𝜕𝑦+ 𝜕𝑦𝑥 ℎ +ℎ + ℎ+ 𝐷 (0) +ℎ

presented in [7], we can further investigate some approximate solutions and stabilities by using the von Neumann condition and the Fourier method. This topic is not further discussed in this paper.

𝜕 , 𝜕ℎ+

4. New Integrable Dynamical Systems

̂8 = 𝜕 + 𝜕 , 𝑋 𝜕𝑦 𝜕𝑦+ pr 𝑋 = 𝑥

𝜕 𝜕 𝜕 𝜕 + 2𝑦 + 𝑥+ + + 2𝑦+ + 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑦

+ ℎ+ 𝐷 (𝑥) +ℎ

𝜕 𝜕 + [𝐷 (2𝑦) − 𝑦1 𝐷 (𝑥)] +ℎ +ℎ 𝜕ℎ+ 𝜕𝑦𝑥 ℎ

+ [𝐷 (𝜁1 ) − 𝑦2 𝐷 (𝑥)] −ℎ

ℎ

ℎ −ℎ

𝜕 , 𝜕𝑦𝑥𝑥 (40)

where 𝑥+ = 𝑆 (𝑥) , +ℎ

4.1. (1 + 1)-Dimensional Integrable Systems. We start with a general loop algebra of the Lie algebra 𝑔4 :

+

𝑦 = 𝑦 (𝑥+ ) , 𝜁1 = 𝐷 (2𝑦) − 𝑦1 𝐷, ℎ

+ℎ

ℎ +ℎ

(41)

𝑦1 = 𝐷 (𝑦) , ℎ

In the section we discuss another application of the Lie algebra 𝑔4 to generate new integrable dynamical systems, including (1 + 1) and (2 + 1) dimensions, according to the Tu and TAH schemes [8, 9]. We have used this approach before to obtain some integrable systems and the corresponding Hamiltonian structures [10–13]. However, we note that the integrable hierarchies derived in this paper possess three potential functions and are different from those in [9– 13]. The integrable dynamical systems here consist of an integrable coupling of the standard Burgers equation, a (2 + 1)-dimensional integrable coupling of the heat conduction equation, and a (2 + 1)-dimensional integrable coupling of a (2 + 1)-dimensional hyperbolic equation.

̃4 = span {𝑋6 (𝑛) , 𝑋7 (𝑛) , 𝑋8 (𝑛) , 𝑋 (𝑛)} , 𝑔 where 𝑋𝑖 (𝑛) = 𝑋𝑖 𝜆𝑛 ,

+ℎ

−ℎ+ℎ

There are a few difference invariants of the Lie algebra 𝑔4 : ℎ+ , ℎ−

𝑐𝑚+1 = −𝑏𝑚𝑥 + 𝑞𝑐𝑚 − 𝑟𝑏𝑚 , (42)

𝑥 , 𝑥

(47)

Taking three constants 𝑏0 = 𝛼, 𝑐0 = 𝛽, and 𝑑0 = 𝛾, we find that (47) is local. For example, we can get 𝑏1 = 𝛾𝑟 − 𝛽𝑠, 𝑐1 = 𝛽𝑞 − 𝛼𝑟, 𝑑1 = 0,

By means of the invariants (42), we can write the following explicit discrete scheme for (26):

ℎ

𝑑𝑚+1 = −𝑐𝑚𝑥 , 𝑏𝑚+1 = 𝑑𝑚𝑥 + 𝑟𝑑𝑚 − 𝑠𝑐𝑚 .

ℎ+ , 𝑥+ 𝑦2 ℎ . 𝑦2

(1−2𝑘)/(1−𝑘) 1 . [𝑘 + (1 − 𝑘) 𝐷 (𝑦)] +ℎ 1 − 2𝑘

(46)

𝑚≥0

+

𝑦2 =

𝜑𝑥 = 𝑈𝜑, 𝑈 = 𝑋 (1) + 𝑞𝑋6 (0) + 𝑟𝑋7 (0) + 𝑠𝑋8 (0) ,

The stationary compatibility condition of (46) leads to

,

ℎ

̃4 , introducing the isospectral problem Now we apply 𝑔

𝑉 = ∑ (𝑏𝑚 𝑋6 (−𝑚) + 𝑐𝑚 𝑋7 (−𝑚) + 𝑑𝑚 𝑋8 (−𝑚)) .

𝑦1 ℎ

(45)

𝜑𝑡 = 𝑉𝜑,

𝑦+ , 𝑦 𝑦2

𝑖 = 6, 7, 8;

𝑋 (𝑛) = 𝑋𝜆𝑛 , 𝑛 ∈ Z.

𝑦2 = 𝐷𝐷 (𝑦) . ℎ

(44)

𝑏2 = −𝛽𝑞𝑠 + 𝛼𝑟𝑠, 2

(43)

This scheme is certainly not unique and one can construct another form of the invariant difference equation. In the way

(48) 2

𝑐2 = −𝛾𝑟𝑥 + 𝛽𝑠𝑥 + 𝛽𝑞 − 𝛼𝑞𝑟 − 𝛾𝑟 + 𝛽𝑟𝑠, 𝑑2 = −𝛽𝑞𝑥 + 𝛼𝑟𝑥 , .. .

6

Advances in Mathematical Physics When one takes 𝑛 = 2, one infers that

Note that 𝑉+(𝑛)

𝑞𝑡2 = 𝑐3

𝑛

= ∑ (𝑏𝑚 𝑋6 (𝑛 − 𝑚) + 𝑐𝑚 𝑋7 (𝑛 − 𝑚) 𝑚=0

(49)

= 𝛽 (𝑞𝑠)𝑥 − 𝛼 (𝑠𝑟)𝑥 − 𝛾𝑞𝑟𝑥 + 𝛽𝑞𝑠𝑥 + 𝛽𝑞3 − 𝛼𝑞2 𝑟 − 𝛾𝑞𝑟2 + 𝛽𝑞𝑟 + 𝛽𝑞𝑟𝑠 − 𝛼𝑠𝑟2 ,

+ 𝑑𝑚 𝑋8 (𝑛 − 𝑚)) = 𝜆𝑛 𝑉 − 𝑉−(𝑛) .

𝑟𝑡2 = 𝑑3

A direct calculation gives

= 𝛾𝑟𝑥𝑥 − 𝛽𝑠𝑥𝑥 − 2𝛽𝑞𝑞𝑥 + 𝛼 (𝑞𝑟)𝑥 + 2𝛾𝑟𝑟𝑥

− (𝑉+(𝑛) )𝑥 + [𝑈, 𝑉+(𝑛) ] = 𝑏𝑛+1 𝑋8 (0) − 𝑐𝑛+1 𝑋6 (0) − 𝑑𝑛+1 𝑋7 (0) .

(56)

− 𝛽 (𝑟𝑠)𝑥 , (50)

𝑠𝑡2 = −𝑏3 = 𝛽𝑞𝑥𝑥 − 𝛼𝑟𝑥𝑥 + 𝛽𝑟𝑞𝑥 − 𝛼𝑟𝑟𝑥 − 𝛾𝑠𝑟𝑥 + 𝛽𝑠𝑠𝑥

According to the Tu scheme, the zero-curvature equation 𝑈𝑡 − (𝑉+(𝑛) )𝑥 + [𝑈, 𝑉+(𝑛) ] = 0

(51)

gives rise to the integrable hierarchy of evolution equations 𝑞

𝑐𝑛+1 (52)

−𝑏𝑛+1

𝑡𝑛

2

𝑞𝑡1 = −𝛾𝑟𝑥 + 𝛽𝑠𝑥 + 𝛽𝑞 − 𝛼𝑞𝑟 − 𝛾𝑟 + 𝛽𝑟𝑠, 𝑟𝑡1 = −𝛽𝑞𝑥 + 𝛼𝑟𝑥 ,

𝑟𝑡 = −𝑠𝑥𝑥 − 2𝑞𝑞𝑥 − (𝑟𝑠)𝑥 ,

(57)

If we let 𝛽 = 𝛾 = 0 and 𝛼 = 1, (56) becomes

(53)

(58)

𝑠𝑡 = −𝑟𝑥𝑥 − 𝑟𝑟𝑥 − 𝑞𝑟𝑠.

𝑞𝑡 = −𝑞𝑟𝑥 − 𝑞𝑟2 , 𝑠𝑡 = −𝑠𝑟𝑥 − 𝑠𝑟2 ,

(59)

𝑟𝑡 = 𝑟𝑥𝑥 + 2𝑟𝑟𝑥 ,

Setting 𝛼 = 𝛾 = 0 and 𝛽 = 1, (53) reduces to 𝑞𝑡 = 𝑠𝑥 + 𝑞2 + 𝑟𝑠, 𝑟𝑡 = −𝑞𝑥 ,

𝑟𝑡 = (𝑞𝑟)𝑥 ,

Setting 𝛼 = 𝛽 = 0 and 𝛾 = 1, (56) gives

𝑠𝑡1 = −𝛽𝑞𝑠 + 𝛼𝑟𝑠.

(54)

𝑠𝑡 = −𝑞𝑠. If we set 𝛾 = 𝛽 = 0 and 𝛼 = 1, then (53) becomes 𝑟𝑡 = 𝑟𝑥 , 𝑞𝑡 = −𝑞𝑟,

𝑞𝑡 = (𝑞𝑠)𝑥 + 𝑞𝑠𝑥 + 𝑞3 + 𝑞𝑟 + 𝑞𝑟𝑠,

𝑞𝑡 = − (𝑟𝑠)𝑥 − 𝑞2 𝑟 − 𝑟2 𝑠,

We consider some reduction cases of (52). When we take 𝑛 = 1, we have 2

Setting 𝛼 = 𝛾 = 0 and 𝛽 = 1, (56) reduces to

𝑠𝑡 = 𝑞𝑥𝑥 + 𝑟𝑞𝑥 + 𝑠𝑠𝑥 + 𝑠𝑞2 + 𝑟𝑠2 .

( 𝑟 ) = ( 𝑑𝑛+1 ) . 𝑠

+ 𝛽𝑠𝑞2 − 𝛼𝑞𝑟𝑠 − 𝛾𝑠𝑟2 + 𝛽𝑟𝑠2 .

(55)

𝑠𝑡 = 𝑟𝑠, which is an integrable coupling of the convective diffusion equation. Equation (55) is solvable.

which is a new (1 + 1)-dimensional integrable coupling of the standard Burgers equation. It is easy to see that when we take 𝑞 = 𝑠 = 0, (59) reduces to the Burgers equation. We can also single out other integrable systems in addition to (53)–(59), but this is not discussed further. 4.2. (2 + 1)-Dimensional Integrable Systems. The Tu scheme is well-known and no review is considered necessary here. This is not the case for the THA scheme, and we briefly discuss how it is used to generate (2 + 1)-dimensional integrable systems [9]. Let A be an associative algebra over the field K = 𝐶 or 𝑅, and assume that 𝜕 : A → A satisfies 𝜕 (𝛼𝑓 + 𝛽𝑔) = 𝛼𝜕𝑓 + 𝛽𝜕𝑔, 𝜕 (𝑓𝑔) = 𝑓 (𝜕𝑔) + (𝜕𝑓) 𝑔, where 𝛼, 𝛽 ∈ K and 𝑓, 𝑔 ∈ A.

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Advances in Mathematical Physics

7

Suppose that A[𝜉] is an associative algebra consisting of 𝑖 the pseudodifferential operators ∑𝑁 𝑖=−∞ 𝑎𝑖 𝜉 , where 𝑎𝑖 ∈ A and 𝜉 satisfies 𝜉𝑓 = 𝑓𝜉 + (𝜕𝑦 𝑓) , 𝑓 ∈ A,

where 𝑏 = ∑ 𝑏𝑚 𝜆−𝑚 , 𝑚≥0

𝑐 = ∑ 𝑐𝑚 𝜆−𝑚 ,

(61)

(70)

𝑚≥0

from which we have

𝑑 = ∑ 𝑑𝑚 𝜆−𝑚 .

𝑛 𝜉𝑛 𝑓 = ∑ ( ) (𝜕𝑖 𝑓) 𝜉𝑛−𝑖 , 𝑖≥0 𝑖

𝑚≥0

Equation (64) leads to

𝑛 𝑓𝜉𝑛 = ∑ (−1)𝑖 ( ) 𝜉𝑛−𝑖 (𝜕𝑖 𝑓) , 𝑖 𝑖≥0

𝑏𝑥 = − (𝜉 + 𝜆) 𝑐 + 𝑞𝑐 − 𝑟𝑏,

(62)

𝑐𝑥 = − (𝜉 + 𝜆) 𝑑, 𝑛 ∈ Z.

(71)

𝑑𝑥 = (𝜉 + 𝜆) 𝑏 − 𝑟𝑑 + 𝑠𝑐. Substituting (70) into (71) yields

Then we take operator matrices,

𝑐𝑚+1 = −𝑏𝑚𝑥 − 𝑐𝑚 𝜉 − 𝑐𝑚𝑦 + 𝑞𝑐𝑚 − 𝑟𝑏𝑚 ,

𝑇

𝑈 = 𝑈 (𝜆, 𝜉, 𝑢) ∈ A [𝜉] , 𝑢 = (𝑢1 , . . . , 𝑢𝑝 ) , 𝑉 = ∑ 𝑉𝑛 𝜆−𝑛 ,

(63)

and seek for solutions of 𝑉 from the stationary zero-curvature equation 𝑉𝑥 = [𝑈, 𝑉] .

𝑑𝑚+1 = −𝑐𝑚𝑥 − 𝑑𝑚 𝜉 − 𝑑𝑚𝑦 ,

(72)

𝑏𝑚+1 = 𝑑𝑚𝑥 − 𝑏𝑚 𝜉 − 𝑏𝑚𝑦 + 𝑟𝑑𝑚 − 𝑠𝑐𝑚 . Assume that 𝑏0 = 𝛼𝜉−1 , 𝑐0 = 𝛽𝜉−1 , 𝑑0 = 𝛿𝜉−1 , where 𝛼, 𝛽, 𝛿 are constants. We can calculate

(64)

𝑏1 = −𝛼 + (𝛿𝑟 − 𝛽𝑠) 𝜉−1 ,

From the expansion

𝑐1 = −𝛽 + (𝛽𝑞 − 𝛼𝑟) 𝜉−1 ,

⟨𝑉,

𝜕𝑈 ⟩ = ∑𝑔𝑖(𝑛) 𝜆−𝑛 , 𝜕𝑢𝑖 𝑖

𝑑1 = −𝛿,

(65)

we obtain a recursion relation among 𝑔(𝑛) = (𝑔1(𝑛) , . . . , 𝑔𝑝(𝑛) )𝑇 , where

𝑏2 = (−𝛿𝑟𝑦 + 𝛽𝑠𝑦 − 𝛽𝑞𝑠 + 𝛼𝑠𝑟) 𝜉−1 + 𝛼𝜉 + 2𝛽𝑠 − 2𝛿𝑟, (73) 𝑐2 = (−𝛿𝑟𝑥 + 𝛽𝑠𝑥 − 𝛽𝑞𝑦 + 𝛼𝑟𝑦 ) 𝜉−1 + 𝛽𝜉 − 𝛽𝑞 + 𝛼𝑟, 𝑑2 = (𝛼𝑟𝑥 − 𝛽𝑞𝑥 ) 𝜉−1 + 𝛿𝜉,

⟨𝑎, 𝑏⟩ = tr 𝑅 (𝑎𝑏) , 𝑅 : A [𝜉] 󳨀→ 𝑅1 , ∑ 𝑎𝑖 𝜉𝑖 󳨀→ 𝑎−1 .

Note that 𝑛

Finally, we try to find an operator 𝐽 from the hierarchy 𝑢𝑡𝑛 = 𝐽𝑔(𝑛) ,

𝑉+(𝑛) = ∑ (𝑏𝑚 𝑋6 (𝑛 − 𝑚) + 𝑐𝑚 𝑋7 (𝑛 − 𝑚) 𝑚=0

(67)

whose Hamiltonian structure can be generated from the (2 + 1)-dimensional trace identity 𝜕 𝜕𝑈 𝛿 ⟨𝑉, 𝑈𝜆 ⟩ = 𝜆−𝛾 𝜆𝛾 ⟨𝑉, ⟩ , 𝑖 = 1, 2, . . . , 𝑝. (68) 𝛿𝑢𝑖 𝜕𝜆 𝜕𝑢𝑖 Based on the above steps for generating (2 + 1)-dimensional integrable hierarchies of evolution equations, we apply the Lie algebra 𝑔4 to introduce a Lax pair for matrices 𝑈 and 𝑉: 𝑈 = (𝜆 + 𝜉) 𝑋 (0) + 𝑞𝑋6 (0) + 𝑟𝑋7 (0) + 𝑠𝑋8 (0) , 𝑉 = 𝑏𝑋6 (0) + 𝑐𝑋7 (0) + 𝑑𝑋8 (0) ,

.. .

(66)

(69)

𝑛

+ 𝑑𝑚 𝑋8 (𝑛 − 𝑚)) = 𝜆 𝑉 −

(74)

𝑉−(𝑛) .

Then (64) can be decomposed into − (𝑉+(𝑛) )𝑥 + [𝑈, 𝑉+(𝑛) ] = (𝑉−(𝑛) )𝑥 − [𝑈, 𝑉−(𝑛) ] .

(75)

The degree of the left-hand side of (75) is ≥0, while the righthand side is ≤0. Therefore, the degrees of both sides are zero. Thus, one infers that − (𝑉+(𝑛) )𝑥 + [𝑈, 𝑉+(𝑛) ] = 𝑅 (𝑏𝑛+1 ) 𝑋8 (0) − 𝑅 (𝑐𝑛+1 ) 𝑋6 (0) − 𝑅 (𝑑𝑛+1 ) 𝑋7 (0) .

(76)

8

Advances in Mathematical Physics

The zero-curvature equation 𝑈𝑡 −

(𝑉+(𝑛) )𝑥

+

[𝑈, 𝑉+(𝑛) ]

=0

(77)

admits the (2 + 1)-dimensional integrable hierarchy 𝑅 (𝑐𝑛+1 )

𝑞

( 𝑟 ) = ( 𝑅 (𝑑𝑛+1 ) ) . 𝑠

𝑡𝑛

(78)

−𝑅 (𝑏𝑛+1 )

When 𝑛 = 1, (78) reduces to 𝑞𝑡 = −𝛿𝑟𝑥 + 𝛽𝑠𝑥 − 𝛽𝑞𝑦 + 𝛼𝑟𝑦 , 𝑟𝑡 = −𝛽𝑞𝑥 + 𝛼𝑟𝑥 ,

(79)

𝑠𝑡 = 𝛿𝑟𝑦 − 𝛽𝑠𝑦 + 𝛽𝑞𝑠 − 𝛼𝑠𝑟. When 𝑛 = 2, (78) reduces to

In the section we want to enlarge the Lie algebra 𝑔4 and deduce the expanding integrable models of the (1 + 1)dimensional integrable hierarchy (52) and the (2 + 1)dimensional integrable hierarchy (78). Obviously, the Lie algebra 𝑔5 is the minimum enlarging Lie algebra of 𝑔4 . However, 𝑔5 cannot generate new integrable dynamical systems based on the isospectral problem (46). That is, by applying the enlarging Lie algebra 𝑔5 we are not able to obtain new expanding integrable models compared to the Lie algebra 𝑔4 , except for an arbitrary smooth function with respect to 𝑥. Therefore, we have to enlarge the Lie algebra 𝑔5 to the following Lie algebra (not unique): ̂ = span {𝑋3 , 𝑋4 } + 𝑔5 , 𝑔

𝑞𝑡 = 𝑅 (𝑐3 ) = 2𝛿𝑟𝑥𝑦 + 𝛽 (𝑞𝑠)𝑥 − 𝛼 (𝑠𝑟)𝑥 + 𝛽𝑞𝑦𝑦 − 𝛼𝑟𝑦𝑦 − 𝛿𝑞𝑟𝑥

(85)

which is also a Lie subalgebra of the large Lie algebra (15). ̂ has the following It is easy to verify that the Lie algebra 𝑔 operation relations:

+ 𝛽𝑞𝑠𝑥 − 𝛽𝑞𝑞𝑦 + 𝛼𝑞𝑟𝑦 + 𝛿𝑟𝑟𝑦 − 𝛽𝑟𝑠𝑦 + 𝛽𝑞𝑟𝑠 − 𝛼𝑠𝑟2 ,

5. Expanding Integrable Models of Integrable Systems (52) and (78)

(80)

𝑟𝑡 = 𝛿𝑟𝑥𝑥 − 𝛽𝑠𝑥𝑥 + 2𝛽𝑞𝑥𝑦 − 2𝛼𝑟𝑥𝑦 ,

[𝑋3 , 𝑋4 ] = −𝑋4 ,

𝑠𝑡 = 𝛽𝑞𝑥𝑥 − 𝛼𝑟𝑥𝑥 𝛿𝑟𝑦𝑦 + 𝛽𝑠𝑦𝑦 − 𝛽 (𝑞𝑠)𝑦 + 𝛼 (𝑠𝑟)𝑦

[𝑋3 , 𝑋5 ] = −𝑋5 , [𝑋3 , 𝑋6 ] = 𝑋6 ,

+ 𝛽𝑟𝑞𝑥 − 𝛼𝑟𝑟𝑥 − 𝛿𝑟𝑥 𝑠 + 𝛽𝑠𝑠𝑥 − 𝛽𝑠𝑞𝑦 + 𝛼𝑠𝑟𝑦 . When 𝛼 = 𝛽 = 0 and 𝛿 = 1, (80) reduces to a (2 + 1)dimensional integrable coupling of the (1 + 1)-dimensional heat conduction equation 𝑟𝑡 = 𝑟𝑥𝑥 , 𝑞𝑡 = 2𝑟𝑥𝑦 − 𝑞𝑟𝑥 + 𝑟𝑟𝑦 ,

(81)

[𝑋4 , 𝑋8 ] = −𝑋5 ,

𝑟𝑡 = −2𝑟𝑥𝑦 , (82)

𝑠𝑡 = −𝑟𝑥𝑥 + (𝑠𝑟)𝑦 − 𝑟𝑟𝑥 + 𝑠𝑟𝑦 , which is a (2 + 1)-dimensional integrable coupling of the (2 + 1)-dimensional hyperbolic equation. When 𝛼 = 𝛿 = 0 and 𝛽 = 1, (80) reduces to

[𝑋4 , 𝑋] = 𝑋3 − 𝑋7 , [𝑋5 , 𝑋6 ] = 𝑋8 , [𝑋5 , 𝑋7 ] = [𝑋5 , 𝑋8 ] = 0, [𝑋5 , 𝑋] = 𝑋5 − 𝑋8 , [𝑋6 , 𝑋7 ] = 𝑋6 ,

𝑟𝑡 = −𝑠𝑥𝑥 + 2𝑞𝑥𝑦 , (83)

[𝑋6 , 𝑋8 ] = 0, [𝑋6 , 𝑋] = −𝑋8 ,

𝑠𝑡 = 𝑞𝑥𝑥 + 𝑠𝑦𝑦 − (𝑞𝑠)𝑦 + 𝑟𝑞𝑥 + 𝑠𝑠𝑥 − 𝑠𝑞𝑦 .

[𝑋7 , 𝑋8 ] = −𝑋8 ,

In particular, when we take 𝑟 = 𝑞 = 0, (83) reduces to 𝑠𝑡 = 𝑠𝑦𝑦 + 𝑠𝑠𝑥 ,

[𝑋4 , 𝑋6 ] = 𝑋7 − 𝑋3 , [𝑋4 , 𝑋7 ] = −𝑋4 ,

When 𝛽 = 𝛿 = 0 and 𝛼 = 1, (80) becomes

𝑞𝑡 = (𝑞𝑠)𝑥 + 𝑞𝑦𝑦 + 𝑞𝑠𝑥 − 𝑞𝑞𝑦 − 𝑟𝑠𝑦 + 𝑞𝑟𝑠,

[𝑋3 , 𝑋] = −𝑋5 − 𝑋6 , [𝑋4 , 𝑋5 ] = 0,

𝑠𝑡 = −𝑟𝑦𝑦 − 𝑟𝑥 𝑠.

𝑞𝑡 = − (𝑠𝑟)𝑥 − 𝑟𝑦𝑦 + 𝑞𝑟𝑦 − 𝑠𝑟2 ,

[𝑋3 , 𝑋7 ] = [𝑋3 , 𝑋8 ] = 0,

(84)

which is a (2 + 1)-dimensional Burgers equation, with (83) being its generalized integrable coupling.

[𝑋7 , 𝑋] = 𝑋6 , [𝑋8 , 𝑋] = 𝑋7 .

(86)

Advances in Mathematical Physics

9

In what follows, we first introduce an isomorphism between ̂; ̂ and the linear space 𝑅7 . Assume that 𝑎, 𝑏 ∈ 𝑔 the Lie algebra 𝑔 then 𝑎, 𝑏 can be expressed by

for 𝑉 is given by 𝑐𝑚+1 = −𝑏𝑚𝑥 + 𝑞𝑐𝑚 + (𝑢1 − 𝑟) 𝑏𝑚 , 𝑑𝑚+1 = −𝑐𝑚𝑥 ,

6

𝑏𝑚+1 + 𝑓𝑚+1 = 𝑑𝑚𝑥 + 𝑞𝑓𝑚 + 𝑟𝑑𝑚 − 𝑠𝑐𝑚 − 𝑢2 𝑏𝑚 ,

𝑎 = ∑𝑎𝑖 𝑋2+𝑖 + 𝑎7 𝑋, 𝑖=1

𝑎𝑚+1 = 𝑞𝑎𝑚 ,

(87)

6

(92)

𝑓𝑚+1 = −𝑓𝑚𝑥 − 𝑢1 𝑓𝑚 + 𝑠𝑎𝑚 ,

𝑏 = ∑𝑏𝑗 𝑋2+𝑗 + 𝑏7 𝑋. 𝑗=1

𝑎𝑚𝑥 = (𝑟 − 𝑢1 ) 𝑎𝑚 .

̂ → 𝑅7 , ∀𝑎 ∈ Suppose that there exists a linear map 𝑓 : 𝑔 𝑇 ̂ → (𝑎1 , . . . , 𝑎7 ) . It is easy to see that 𝑓 is an isomorphism 𝑔 ̂ and 𝑅7 . Based on the commutabetween the linear spaces 𝑔 ̂ , we define an operation in tive relations of the Lie algebra 𝑔 𝑅7 by

Taking 𝑏0 = 𝛼, 𝑐0 = 𝛽, 𝑑0 = 𝛾, 𝑎0 = 𝜎,

[𝑎, 𝑏]𝑅 = (𝑎4 𝑏2 − 𝑎2 𝑏4 + 𝑎2 𝑏7 − 𝑎7 𝑏2 , 𝑎2 𝑏1 − 𝑎1 𝑏2

𝑓0 = 𝛿,

+ 𝑎5 𝑏2 − 𝑎2 𝑏5 , 𝑎3 𝑏1 − 𝑎1 𝑏3 + 𝑎3 𝑏7 − 𝑎7 𝑏3 + 𝑎6 𝑏2 − 𝑎2 𝑏6 + 𝑎7 𝑏1 − 𝑎1 𝑏7 , 𝑎1 𝑏4 − 𝑎4 𝑏1 + 𝑎4 𝑏5 − 𝑎5 𝑏4 + 𝑎5 𝑏7 − 𝑎7 𝑏5 , 𝑎2 𝑏4 − 𝑎4 𝑏2 + 𝑎7 𝑏2 − 𝑎2 𝑏7 + 𝑎6 𝑏7

(93)

then (92) admits that 𝑎1 = 𝜎𝑞,

(88)

𝑏1 = 𝛿𝑞 + 𝛾𝑟 − 𝛽𝑠 − 𝛼𝑢2 ,

− 𝑎7 𝑏6 + 𝑎7 𝑏1 − 𝑎1 𝑏7 , 𝑎3 𝑏4 − 𝑎4 𝑏3 + 𝑎7 𝑏4 − 𝑎4 𝑏7

𝑐1 = 𝛽𝑞 + 𝛼 (𝑢1 − 𝑟) ,

𝑇

+ 𝑎6 𝑏5 − 𝑎5 𝑏6 + 𝑎7 𝑏3 − 𝑎3 𝑏7 , 0) .

(94)

𝑑1 = 0,

We can prove that the linear space 𝑅7 becomes a Lie algebra if it is equipped with the operation (88). Next, we deduce the expanding integrable models of the integrable hierarchies based on the Lie algebra 𝑅7 . 5.1. Expanding Integrable Model of the Integrable Hierarchy (52). We introduce a loop algebra ̃7 = {(𝑥 , . . . , 𝑥 , 0)𝑇 , 𝑥 = ∑ 𝑥 𝜆𝑚 , 𝑥 𝑅 1 6 𝑖 𝑖𝑚 𝑖𝑚

𝑓1 = −𝛿𝑢1 + 𝜎𝑠, .. . Note that 𝑛

𝑇

𝑉+(𝑛) = ∑ (0, 𝑎𝑚 , 𝑓𝑚 , 𝑏𝑚 , 𝑐𝑚 , 𝑑𝑚 , 0) 𝜆𝑛−𝑚 𝑚=0

(95)

= 𝜆𝑛 𝑉 − 𝑉−(𝑛) . (89)

= 𝑥𝑖𝑚 (𝑡, 𝑥) , 𝑖 = 1, . . . , 6} .

A direct calculation yields − (𝑉+(𝑛) )𝑥 + [𝑈, 𝑉+(𝑛) ]𝑅 = (−𝑎𝑛+1 , 0, −𝑓𝑛+1 , 𝑇

(96)

− 𝑐𝑛+1 , 𝑎𝑛+1 − 𝑑𝑛+1 , 𝑏𝑛+1 + 𝑓𝑛+1 , 0) .

̃7 , we consider the Lax matrices In terms of 𝑅

Thus the zero-curvature equation 𝑇

𝑈 = (𝑢1 , 0, 𝑢2 , 𝑞, 𝑟, 𝑠, 𝜆) , 𝑇

(90)

𝑉 = (0, 𝑎, 𝑓, 𝑏, 𝑐, 𝑑, 0) ,

𝑈𝑡𝑛 − (𝑉+(𝑛) )𝑥 + [𝑈, 𝑉+(𝑛) ]𝑅 = 0 has an integrable hierarchy

where 𝜆 is a spectral parameter, 𝑎 = ∑𝑚≥0 𝑎𝑚 𝜆−𝑚 , 𝑏 = ∑𝑚≥0 𝑏𝑚 𝜆−𝑚 , . . .. According to the Tu scheme, a solution to the stationary zero-curvature equation

𝑐𝑛+1

𝑞

𝑟 𝑑𝑛+1 − 𝑎𝑛+1 (−𝑏 − 𝑓 ) (𝑠) ( ) = ( 𝑛+1 𝑛+1 ) . 𝑢1

𝑉𝑥 = [𝑈, 𝑉]𝑅

(97)

(91)

(𝑢2 )𝑡𝑛

𝑎𝑛+1 (

𝑓𝑛+1

)

(98)

10

Advances in Mathematical Physics

When we take 𝑢1 = 𝑢2 = 0, (98) reduces to the (1 + 1)-dimensional integrable hierarchy (52). Therefore, (98) is an expanding integrable model of (52). A simple reduction of (98), when 𝑛 = 1, has the form

𝑏𝑥 = − (𝜆 + 𝜉) 𝑐 + 𝑞𝑐 + (𝑢 − 𝑟) 𝑏, 𝑐𝑥 = − (𝜆 + 𝜉) 𝑑 + (𝜆 + 𝜉) 𝑎 − 𝑞𝑎,

𝑞𝑡 = 𝑐2

𝑑𝑥 = (𝜆 + 𝜉) 𝑏 + (𝜆 + 𝜉) 𝑓 − 𝑞𝑓 − 𝑟𝑑 + 𝑠𝑐 + V𝑏,

= −𝛿𝑞𝑥 − 𝛾𝑟𝑥 + 𝛽𝑠𝑥 + 𝛼𝑢2𝑥 + 𝛽𝑞2 − 𝛼𝑞𝑟 + 2𝛿𝑞

𝑎𝑥 = (𝑟 − 𝑢) 𝑎,

1 + (2𝛾𝑟𝑞𝑥 − 2𝛽𝑞𝑥 𝑠 − 2𝛼𝑞𝑥 𝑢2 ) , 𝑞

(105)

𝑓𝑥 = − (𝜆 + 𝜉) 𝑓 + 𝑠𝑎 − 𝑢𝑓,

𝑟𝑡 = −𝛽𝑞𝑥 − 𝛼𝑟𝑥 − 𝜎𝑞2 ,

(99)

𝑠𝑡 = 𝛿𝑞𝑢1 − 𝜎𝑞𝑠 + 𝛽𝑞𝑠 − 𝛼𝑠𝑟 + 𝛿𝑞𝑢2 + 𝛾𝑢2 𝑟 − 𝛽𝑠𝑢2 −

Similar to previous discussions, there exist relations among 𝑎, 𝑏, 𝑐, 𝑑, 𝑓 such that

0 = − (𝜆 + 𝜉) 𝑎 + 𝑞𝑑. Substituting (104) into (105) gives rise to

𝛼𝑢22 ,

𝑐𝑚+1 = −𝑏𝑚𝑥 − 𝑐𝑚 𝜉 − 𝑐𝑚𝑦 + 𝑞𝑐𝑚 + (𝑢 − 𝑟) 𝑏𝑚 ,

2

𝑢1,𝑡 = 𝜎𝑞 ,

𝑑𝑚+1 = −𝑐𝑚𝑥 − 𝑑𝑚 𝜉 − 𝑑𝑚𝑦 ,

𝑢2,𝑡 = 𝛿𝑢1𝑥 + 𝜎𝑠𝑥 +

𝛿𝑢12

− 𝜎𝑢1 𝑠 + 𝜎𝑞𝑠.

𝑏𝑚+1 + 𝑓𝑚+1 = 𝑑𝑚𝑥 − 𝑏𝑚 𝜉 − 𝑏𝑚𝑦 − 𝑓𝑚 𝜉 − 𝑓𝑚𝑦 + 𝑞𝑓𝑚 + 𝑟𝑑𝑚 − 𝑠𝑐𝑚 − V𝑏𝑚 ,

5.2. Expanding Integrable Model of the (2 + 1)-Dimensional Integrable Hierarchy (78). We define an associative algebra 𝑅7 [𝜉] consisting of elements ∑ 𝑎𝑖 𝜉𝑖 , 𝑎𝑖 ∈ 𝑅7 . A linear operator on 𝑅7 [𝜉] is given by 𝜕 : 𝑅7 [𝜉] 󳨀→ 𝑅7 [𝜉] ,

(106)

𝑓𝑚+1 = −𝑓𝑚𝑥 − 𝑓𝑚 𝜉 − 𝑓𝑚𝑦 + 𝑠𝑎𝑚 − 𝑢𝑓𝑚 , 𝑎𝑚+1 = 𝑞𝑎𝑚 − 𝑎𝑚 𝜉 − 𝑎𝑚𝑦 , 𝑎𝑚𝑥 = (𝑟 − 𝑢) 𝑎𝑚 .

(100) Taking

which satisfies 𝜕 (𝑓𝑔) = (𝜕𝑓) 𝑔 + 𝑓 (𝜕𝑔) , 𝜕 (𝛼𝑓 + 𝛽𝑔) = 𝛼 (𝜕𝑓) + 𝛽 (𝜕𝑔) ,

𝑏0 = 𝛼𝜉−1 ,

(101)

𝑐0 = 𝛽𝜉−1 ,

where 𝑓, 𝑔 ∈ 𝑅7 [𝜉] and 𝛼, 𝛽 are arbitrary constants. In addition, a residue operator is defined by

𝑑0 = 𝛿𝜉−1 ,

R : 𝑅7 [𝜉] 󳨀→ 𝑅1 ,

𝑓0 = 𝛾𝜉−1 ,

(102)

R (∑ 𝑎𝑖 𝜉𝑖 ) = 𝑎−1 .

we have from (106) that

Based on the above notations, we introduce two Lax matrices on 𝑅7 [𝜉]: 𝑇

𝑈 = (𝑢, 0, V, 𝑞, 𝑟, 𝑠, 𝜆 + 𝜉) , 𝑇

𝑎0 = 𝜎𝜉−1 ,

(103)

𝑉 = (0, 𝑎, 𝑓, 𝑏, 𝑐, 𝑑, 0) ,

𝑎1 = −𝜎 + 𝜎𝑞𝜉−1 , 𝑓1 = 𝛾 + (𝜎𝑠 − 𝛾𝑢) 𝜉−1 , 𝑑1 = −𝛿, 𝑐1 = −𝛽 + 𝛽𝑞𝜉−1 , 𝑏1 + 𝑓1 = −𝛼 − 𝛾 + (𝛾𝑞 + 𝛿𝑟 − 𝛽𝑠 − 𝛼V) 𝜉−1 ,

where

𝑏1 = −𝛼 − 2𝛾 + (𝛾𝑞 + 𝛿𝑟 − 𝛽𝑠 − 𝛼V − 𝜎𝑠 + 𝛾𝑢) 𝜉−1 ,

𝑎 = ∑ 𝜆−𝑚 , 𝑚≥0

𝑏 = ∑ 𝑏𝑚 𝜆−𝑚 , 𝑚≥0

.. .

𝑐2 = −𝛽𝑞 + (𝛼 + 2𝛾) (𝑟 − 𝑢) + [−𝛾𝑞𝑥 − 𝛿𝑟𝑥 + 𝛽𝑠𝑥 (104)

+ 𝛼V𝑥 + 𝜎𝑠𝑥 − 𝛾𝑢𝑥 − 𝛽𝑞𝑦 + 𝛽𝑞2 + (𝑢 − 𝑟) (𝛾𝑞 + 𝛿𝑟 − 𝛽𝑠 − 𝛼V − 𝜎𝑠 + 𝛾𝑢)] 𝜉−1 ,

(107)

Advances in Mathematical Physics

11

𝑑2 = −𝛿𝜉 − 𝛽𝑞𝑥 𝜉−1 ,

and the (1+1)-dimensional and (2+1)-dimensional integrable dynamical hierarchies as well as their expanding integrable models. We have adopted the vector Lie algebra 𝑅7 to deduce the expanding integrable models of the integrable hierarchies, so that we can generate Hamiltonian structures of the expanding integrable models by the quadratic-form identity or the variational identity in [14, 15]. It is worth further studying discretizations of the differential equations obtained in this paper. By applying some schemes in [16], we can further investigate the discretizations’ exact solutions, symmetries, infinitely many conservation laws, and quasiperiodic solutions by following the ways in [17–32].

𝑓2 = −𝛾𝜉 − 2𝜎𝑠 + (−𝜎𝑠𝑥 + 𝛾𝑢𝑥 − 𝜎𝑠𝑦 + 𝛾𝑢𝑦 + 𝜎𝑞𝑠 − 𝜎𝑢𝑠 + 𝛾𝑢2 ) 𝜉−1 , 𝑏2 + 𝑓2 = (𝛼 + 𝛾) 𝜉 + 2𝛾𝑞 − 𝜎𝑠 + 𝛾𝑢 + 𝛾V + (−𝛾𝑞𝑦 − 𝛿𝑟𝑦 + 𝛽𝑠𝑦 + 𝛼V𝑦 − 𝛾𝑞𝑢 − 𝛽𝑞𝑠 − 𝛾𝑞V − 𝛿𝑟V + 𝛽V𝑠 + 𝛼V2 ) 𝜉−1 , 𝑎2 = −𝜎𝑞 + 𝜎 (𝑞2 − 𝑞𝑦 ) 𝜉−1 , (108) 𝑞𝑥 = 𝑞 (𝑟 − 𝑢) ,

(109)

(𝑞2 − 𝑞𝑦 )𝑥 = 𝑞2 − 𝑞𝑦 .

R (𝑐𝑛+1 )

R (𝑑𝑛+1 − 𝑎𝑛+1 ) 𝑟 ( (𝑠) ) ( ) = (−R (𝑏𝑛+1 + 𝑓𝑛+1 )) . 𝑢 ( V )𝑡𝑛

(110)

R (𝑎𝑛+1 ) (

R (𝑓𝑛+1 )

The authors declare that they have no competing interests.

Acknowledgments

According to the TAH scheme, we form a (2+1)-dimensional integrable hierarchy 𝑞

Competing Interests

)

When we set 𝑢 = V = 0, (110) reduces to the (2 + 1)-dimensional integrable hierarchy (78). Hence, (110) is an expanding integrable model of (78). A simple reduction of (110), when 𝑛 = 1, has the form 𝑞𝑡 = R (𝑐2 ) = −𝛾𝑞𝑥 − 𝛿𝑟𝑥 + 𝛽𝑠𝑥 + 𝛼V𝑥 + 𝜎𝑠𝑥 − 𝛾𝑢𝑥 − 𝛽𝑞𝑦 + 𝛽𝑞2 + (𝑢 − 𝑟) (𝛾𝑞 + 𝛿𝑟 − 𝛽𝑠 − 𝛼V − 𝜎𝑠 + 𝛾𝑢) , 𝑟𝑡 = −𝛽𝑞𝑥 + 𝜎 (𝑞𝑦 − 𝑞2 ) , 𝑠𝑡 = 𝛾𝑞𝑦 + 𝛿𝑟𝑦 − 𝛽𝑠𝑦 − 𝛼V𝑦 + 𝛾𝑞𝑢 + 𝛽𝑞𝑠 + 𝛾𝑞V + 𝛿𝑟V

(111)

− 𝛽V𝑠 − 𝛼V2 , 𝑢𝑡 = 𝜎 (𝑞2 − 𝑞𝑦 ) , V𝑡 = −𝜎𝑠𝑥 + 𝛾𝑢𝑥 − 𝜎𝑠𝑦 + 𝛾𝑢𝑦 + 𝜎𝑞𝑠 − 𝜎𝑢𝑠 + 𝛾𝑢2 , which has the constraint condition (109). If we further take 𝑛 = 2, 3, . . ., we obtain different (2+1)-dimensional integrable systems with their corresponding constraint conditions.

6. Conclusions In the paper we have discussed two applications of the operator Lie algebra (15) and derived some new results. These include a second-order differential equation and its Lagrangians, the quasi-extremal equation and its general solutions,

This work was supported by the National Natural Science Foundation of China (Grant no. 11371361), the Innovation Team of Jiangsu Province hosted by China University of Mining and Technology (2014) and the Fundamental Research Funds for the Central Universities (2013XK03), the Natural Science Foundation of Shandong Province (Grant no. ZR2013AL016), Hong Kong Research Grant Council (Grant no. HKBU202512), and Key Discipline Construction by China University of Mining and Technology (Grant no. XZD201602).

References [1] D. Levi and P. Winternitz, “Symmetries and conditional symmetries of differential-difference equations,” Journal of Mathematical Physics, vol. 34, no. 8, pp. 3713–3730, 1993. [2] D. Levi and P. Winternitz, “Symmetries of discrete dynamical systems,” Journal of Mathematical Physics, vol. 37, no. 11, pp. 5551–5576, 1996. [3] M. I. Bakirova, V. A. Dorodnitsyn, and R. V. Kozlov, “Symmetry-preserving difference schemes for some heat transfer equations,” Journal of Physics A: Mathematical and General, vol. 30, no. 23, pp. 8139–8155, 1997. [4] V. Dorodnitsyn, R. Kozlov, and P. Winternitz, “Continuous symmetries of Lagrangians and exact solutions of discrete equations,” Journal of Mathematical Physics, vol. 45, no. 1, pp. 336– 359, 2004. [5] V. Dorodnitsyn, “Noether-type theorems for difference equations,” Applied Numerical Mathematics, vol. 39, no. 3-4, pp. 307– 321, 2001. [6] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, vol. 81 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989. [7] Y.-F. Zhang, L.-X. Wu, and W.-J. Rui, “A corresponding Lie algebra of a reductive homogeneous group and its applications,” Communications in Theoretical Physics, vol. 63, no. 5, pp. 535– 548, 2015. [8] G. Z. Tu, “The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems,” Journal of Mathematical Physics, vol. 30, no. 2, pp. 330–338, 1989.

12 [9] G. Z. Tu, R. I. Andrushkiw, and X. C. Huang, “A trace identity and its application to integrable systems of 1 + 2 dimensions,” Journal of Mathematical Physics, vol. 32, no. 7, pp. 1900–1907, 1991. [10] Y. Zhang and W. Rui, “On generating (2+1)-dimensional hierarchies of evolution equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 10, pp. 3454–3461, 2014. [11] Y. F. Zhang and W. J. Rui, “A few super-integrable hierarchies and some reductions, super-Hamiltonian structures,” Reports on Mathematical Physics, vol. 75, no. 2, pp. 231–255, 2015. [12] Y. F. Zhang, H. W. Tam, and J. Q. Mei, “Some 2+1 dimensional super-integrable systems,” Zeitschrift f¨ur Naturforschung A, vol. 70, no. 10, pp. 791–796, 2015. [13] Y. Zhang and L. Wu, “Two (2+1)-dimensional expanding dynamical systems associated to the mKP hierarchy,” Applied Mathematics and Computation, vol. 268, pp. 561–574, 2015. [14] F. K. Guo and Y. F. Zhang, “The quadratic-form identity for constructing the Hamiltonian structure of integrable systems,” Journal of Physics A: Mathematical and General, vol. 38, no. 40, pp. 8537–8548, 2005. [15] W.-X. Ma and M. Chen, “Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras,” Journal of Physics A: Mathematical and General, vol. 39, no. 34, pp. 10787–10801, 2006. [16] S. Y. Lou, “A note on the new similarity reductions of the Boussinesq equation,” Physics Letters A, vol. 151, no. 3-4, pp. 133– 135, 1990. [17] S. Y. Lou, “Generalized symmetries and 𝑊∞ algebras in threedimensional Toda field theory,” Physical Review Letters, vol. 71, no. 25, pp. 4099–4102, 1993. [18] S. Y. Lou, “Symmetries of the Kadomtsev-Petviashvili equation,” Journal of Physics A: Mathematical and General, vol. 26, no. 17, pp. 4387–4394, 1993. [19] C. Cao, X. Geng, and Y. Wu, “From the special 2+1 Toda lattice to the Kadomtsev-Petviashvili equation,” Journal of Physics. A. Mathematical and General, vol. 32, no. 46, pp. 8059–8078, 1999. [20] C.-W. Cao and X.-X. Xu, “A finite genus solution of the Veselov’s discrete Neumann system,” Communications in Theoretical Physics, vol. 58, no. 4, pp. 469–474, 2012. [21] C. Cao and X. Yang, “A (2+1)-dimensional derivative Toda equation in the context of the Kaup-Newell spectral problem,” Journal of Physics. A. Mathematical and Theoretical, vol. 41, no. 2, Article ID 025203, 19 pages, 2008. [22] X. G. Geng, H. H. Dai, and J. Y. Zhu, “Decomposition of the discrete Ablowitz-Ladik hierarchy,” Studies in Applied Mathematics, vol. 118, no. 3, pp. 281–312, 2007. [23] X. G. Geng, H. H. Dai, and C. Cao, “Algebro-geometric constructions of the discrete Ablowitz-Ladik flows and applications,” Journal of Mathematical Physics, vol. 44, no. 10, pp. 4573– 4588, 2003. [24] E. G. Fan and Z. H. Yang, “A lattice hierarchy with a free function and its reductions to the Ablowitz-Ladik and Volterra hierarchies,” International Journal of Theoretical Physics, vol. 48, no. 1, pp. 1–9, 2009. [25] Z.-N. Zhu and H.-W. Tam, “Nonisospectral negative Volterra flows and mixed Volterra flows: lax pairs, infinitely many conservation laws and integrable time discretization,” Journal of Physics. A. Mathematical and General, vol. 37, no. 9, pp. 3175– 3187, 2004.

Advances in Mathematical Physics [26] D. J. Zhang and D. Y. Chen, “Hamiltonian structure of discrete soliton systems,” Journal of Physics A: Mathematical and General, vol. 35, no. 33, pp. 7225–7241, 2002. [27] Y. B. Suris, “On some integrable systems related to the Toda lattice,” Journal of Physics A: Mathematical and General, vol. 30, no. 6, pp. 2235–2249, 1997. [28] W.-X. Ma and B. Fuchssteiner, “Algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations,” Journal of Mathematical Physics, vol. 40, no. 5, pp. 2400–2418, 1999. [29] Z. J. Qiao, “𝑟-matrix and algebraic-geometric solution for integrable symplectic map,” Chinese Science Bulletin, vol. 44, no. 2, pp. 114–118, 1999. [30] Z. Zhu, H. Huang, and W. Xue, “Lax representations and integrable time discretizations of the DDKdV, DDmKdV, and DDHOKdV,” Physics Letters. A, vol. 252, no. 3-4, pp. 180–190, 1999. [31] Z.-N. Zhu, H.-Q. Zhao, and F.-F. Zhang, “Explicit solutions to an integrable lattice,” Studies in Applied Mathematics, vol. 125, no. 1, pp. 55–67, 2010. [32] W.-X. Ma, “Conservation laws of discrete evolution equations by symmetries and adjoint symmetries,” Symmetry, vol. 7, no. 2, pp. 714–725, 2015.

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