New Applications of a Kind of Infinitesimal-Operator Lie Algebra

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 () and () dimensions.


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][4][5].We first briefly recall some fundamental notations.Given  satisfying the Euler-Lagrange equation, where is a Lie group operator, it is well-known that the functional achieves its extremal values on [1].Equation ( 1) is an ODE that can be rewritten as Group  generated by the vector field  =  (, )   +  (, )   (5) is a variational symmetry of the functional L if and only if the Lagrangian satisfies pr  () +  () = 0.
Therefore, if the vector operator  is a divergence symmetry, we have a first integral 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 = ∑ Ω L (,  + , ,  + ) ℎ + (10) 2 Advances in Mathematical Physics defined on some one-dimensional lattice Ω with step spacing ℎ + , which can be represented by ℎ + =  (, ,  + ,  + ) .(11) For an arbitrary curve the stationary value of a differential functional is given by any solution of the quasiextremal equations [4,5]: It can be verified that if the Lagrangian density L is divergence invariant under group , it holds that then each element  of the Lie algebra corresponding to  provides us with a first integral of ( 12): In the paper, we start from the infinitesimal symmetry operators of an eight-parameter projective transformation group in  2 [6], 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 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.

A Few Lie Subalgebras of the Operator Algebra
The Lie algebra (15) has the following commutative operations [6]: From this Lie algebra, some interesting Lie algebras given in [4] can be obtained.In fact, the Lie algebras in [4] denoted by (17) 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 then it is easy to have where  1 and  2 are Lie subalgebras of  and both are not semisimple.In addition, we have Hence, the Lie algebra  is not a direct sum of the Lie subalgebras  1 and  2 .Because of relation (20),  is not a symmetric Lie algebra.Therefore,  itself cannot be utilized to generate integrable couplings.However, subalgebras of  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  3 ∈  1 and  5 ,  6 ,  7 ∈  2 to make a linear combination: It is easy to verify that If we let  = −1 and remove the element  5 from the Lie algebra  5 , we get a 4-dimensional Lie subalgebra The corresponding commutative operations are given by 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.

General Solutions of Some Difference Equations
A direct calculation yields a second-order ODE corresponding to the vector field  on the space (, ,   ,   ): This equation can be obtained from the Lagrangian Obviously, the Lagrangian admits the symmetries  5 and  8 : pr  5 () +  ( 5 ) = 0, With the help of Noether's theorem we obtain the following first integrals: Solving the second equation for   in (29) and substituting it into the first equation, we obtain which is the general solution to (26).
Let us take the difference Lagrangians which satisfies pr where Then the variations of L yield the quasiextremal equations where 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: Solving   from the second equation in (37) and then substituting the solution into the third one yields the lattice equation: Inserting   into the first equation in (37) gives rise to the general solution to the quasiextremal equation (35): 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) where There are a few difference invariants of the Lie algebra  4 : By means of the invariants (42), we can write the following explicit discrete scheme for (26): This scheme is certainly not unique and one can construct another form of the invariant difference equation.In the way 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.

New Integrable Dynamical Systems
In the section we discuss another application of the Lie algebra  4 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][11][12][13].However, we note that the integrable hierarchies derived in this paper possess three potential functions and are different from those in [9][10][11][12][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.

4.1.
(1 + 1)-Dimensional Integrable Systems.We start with a general loop algebra of the Lie algebra  4 : The stationary compatibility condition of (46) leads to Taking three constants  0 = ,  0 = , and  0 = , we find that (47) is local.For example, we can get Advances in Mathematical Physics Note that A direct calculation gives According to the Tu scheme, the zero-curvature equation gives rise to the integrable hierarchy of evolution equations We consider some reduction cases of (52).When we take  = 1, we have Setting  =  = 0 and  = 1, (53) reduces to If we set  =  = 0 and  = 1, then (53) becomes which is an integrable coupling of the convective diffusion equation.Equation ( 55) is solvable.
When one takes  = 2, one infers that Setting  =  = 0 and  = 1, (56) reduces to If we let  =  = 0 and  = 1, (56) becomes Setting  =  = 0 and  = 1, (56) gives which is a new (1 + 1)-dimensional integrable coupling of the standard Burgers equation.It is easy to see that when we take  =  = 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.

(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 =  or , and assume that  : A → A satisfies where ,  ∈ K and ,  ∈ A.
Suppose that A[] is an associative algebra consisting of the pseudodifferential operators ∑  =−∞     , where   ∈ A and  satisfies from which we have Then we take operator matrices, and seek for solutions of  from the stationary zero-curvature equation From the expansion we obtain a recursion relation among  () = ( () 1 , . . .,  ()  )  , where Finally, we try to find an operator  from the hierarchy whose Hamiltonian structure can be generated from the (2 + 1)-dimensional trace identity Based on the above steps for generating (2 + 1)-dimensional integrable hierarchies of evolution equations, we apply the Lie algebra  4 to introduce a Lax pair for matrices  and :  = ( + )  (0) +  6 (0) +  7 (0) +  8 (0) , where Equation ( 64) leads to Substituting ( 70) into (71) yields Assume that  0 =  −1 ,  0 =  −1 ,  0 =  −1 , where , ,  are constants.We can calculate Note that Then (64) can be decomposed into 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 When  =  = 0 and  = 1, (80) becomes In particular, when we take  =  = 0, (83) reduces to which is a (2 + 1)-dimensional Burgers equation, with (83) being its generalized integrable coupling.

Expanding Integrable Models of Integrable Systems (52) and (78)
In the section we want to enlarge the Lie algebra  4 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  5 is the minimum enlarging Lie algebra of  4 .However,  5 cannot generate new integrable dynamical systems based on the isospectral problem (46).That is, by applying the enlarging Lie algebra  5 we are not able to obtain new expanding integrable models compared to the Lie algebra  4 , except for an arbitrary smooth function with respect to .Therefore, we have to enlarge the Lie algebra  5 to the following Lie algebra (not unique): which is also a Lie subalgebra of the large Lie algebra (15).It is easy to verify that the Lie algebra ĝ has the following operation relations: In what follows, we first introduce an isomorphism between the Lie algebra ĝ and the linear space  7 .Assume that ,  ∈ ĝ; then ,  can be expressed by Suppose that there exists a linear map  : ĝ → We can prove that the linear space  7 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  7 .) )