Research Article | Open Access
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.
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 satisfying the Euler-Lagrange equation, whereis a Lie group operator, it is well-known that the functional achieves its extremal values on .
Equation (1) is an ODE that can be rewritten as Group generated by the vector field is a variational symmetry of the functional if and only if the Lagrangian satisfies If there exists a function such that then the group is known as an infinitesimal divergence symmetry. By Noether’s theorem, one infers that  Therefore, if the vector operator is a divergence symmetry, we have a first integral which corresponds to the Euler-Lagrange equation.
In , the Lagrangian formalism for second-order difference equations was reviewed. Consider a finite difference functional defined on some one-dimensional lattice with step spacing , which can be represented by 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 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 , to discuss some continuous symmetries of Lagrangians and general solutions of discrete equations. We generate new ()-dimensional and ()-dimensional integrable systems with three potential functions. Specifically, we obtain an integrable coupling of the standard Burgers equation and a ()-dimensional integrable coupling of the heat conduct equation. We also derive a ()-dimensional integrable coupling of the ()-dimensional hyperbolic equation under the framework of the Tu scheme. Finally, we establish a vector Lie algebra to generate expanding integrable models of the ()-dimensional and ()-dimensional integrable hierarchies of evolution equations obtained in the paper.
2. A Few Lie Subalgebras of the Operator Algebra
The Lie algebra (15) has the following commutative operations :From this Lie algebra, some interesting Lie algebras given in  can be obtained. In fact, the Lie algebras in  denoted by are actually linear combinations of the elements in (15): In , 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 and 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 and . 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 and to make a linear combination:It is easy to verify that is a Lie subalgebra of , with the commutative relationsIf we let and remove the element from the Lie algebra , we get a 4-dimensional Lie subalgebra The corresponding commutative operations are given byIn 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 ()-dimensional and ()-dimensional integrable hierarchies of evolution equations.
3. 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 and : 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 where Then the variations of yield the quasiextremal equations where and are spatial spacing steps. The case 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) admits a Lie transformation group that can be represented by the Lie algebra . The group consists of infinitesimal vector fields. The corresponding prolongation operators are given bywhereThere are a few difference invariants of the Lie algebra : 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 , 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.
4. New Integrable Dynamical Systems
In the section we discuss another application of the Lie algebra to generate new integrable dynamical systems, including () and () 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 ()-dimensional integrable coupling of the heat conduction equation, and a ()-dimensional integrable coupling of a ()-dimensional hyperbolic equation.
4.1. )-Dimensional Integrable Systems
We start with a general loop algebra of the Lie algebra : whereNow we apply , introducing the isospectral problem The stationary compatibility condition of (46) leads to Taking three constants , , and , we find that (47) is local. For example, we can getNote 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 , we have Setting and , (53) reduces to If we set and , then (53) becomes which is an integrable coupling of the convective diffusion equation. Equation (55) is solvable.
When one takes , one infers that Setting and , (56) reduces to If we let and , (56) becomes Setting and , (56) gives which is a new ()-dimensional integrable coupling of the standard Burgers equation. It is easy to see that when we take , (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. ()-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 ()-dimensional integrable systems . Let be an associative algebra over the field or , and assume that satisfies where and
Suppose that is an associative algebra consisting of the pseudodifferential operators , where 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 , where Finally, we try to find an operator from the hierarchy whose Hamiltonian structure can be generated from the ()-dimensional trace identity Based on the above steps for generating ()-dimensional integrable hierarchies of evolution equations, we apply the Lie algebra to introduce a Lax pair for matrices and : where Equation (64) leads to Substituting (70) into (71) yields Assume that , , , where , , are constants. We can calculateNote that Then (64) can be decomposed into The degree of the left-hand side of (75) is ≥0, while the right-hand side is ≤0. Therefore, the degrees of both sides are zero. Thus, one infers that The zero-curvature equation admits the ()-dimensional integrable hierarchy When , (78) reduces to When , (78) reduces to When and , (80) reduces to a ()-dimensional integrable coupling of the ()-dimensional heat conduction equation When and , (80) becomes which is a ()-dimensional integrable coupling of the ()-dimensional hyperbolic equation.
In the section we want to enlarge the Lie algebra and deduce the expanding integrable models of the ()-dimensional integrable hierarchy (52) and the ()-dimensional integrable hierarchy (78). Obviously, the Lie algebra is the minimum enlarging Lie algebra of . However, cannot generate new integrable dynamical systems based on the isospectral problem (46). That is, by applying the enlarging Lie algebra we are not able to obtain new expanding integrable models compared to the Lie algebra , except for an arbitrary smooth function with respect to . Therefore, we have to enlarge the Lie algebra 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 has the following operation relations:In what follows, we first introduce an isomorphism between the Lie algebra and the linear space . Assume that ; then can be expressed by Suppose that there exists a linear map It is easy to see that is an isomorphism between the linear spaces and . Based on the commutative relations of the Lie algebra , we define an operation in by We can prove that the linear space 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 .
5.1. Expanding Integrable Model of the Integrable Hierarchy (52)
We introduce a loop algebra In terms of , we consider the Lax matrices where is a spectral parameter, . According to the Tu scheme, a solution to the stationary zero-curvature equation for is given by Taking then (92) admits thatNote that A direct calculation yields Thus the zero-curvature equation has an integrable hierarchy When we take , (98) reduces to the ()-dimensional integrable hierarchy (52). Therefore, (98) is an expanding integrable model of (52). A simple reduction of (98), when , has the form
5.2. Expanding Integrable Model of the ()-Dimensional Integrable Hierarchy (78)
We define an associative algebra consisting of elements , A linear operator on is given by which satisfies where and are arbitrary constants. In addition, a residue operator is defined by Based on the above notations, we introduce two Lax matrices on : where Similar to previous discussions, there exist relations among such that Substituting (104) into (105) gives rise to Taking we have from (106) that