Research Article | Open Access
Two Expanding Integrable Models of the Geng-Cao Hierarchy
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.
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 , the coupled variable satisfying is linear in . The reason for this may be given by special Lie algebras. That is, such a Lie algebra can be decomposed into a sum of the two subalgebras and , which meets
If the subalgebra is not simple, then the integrable coupling is linear with respect to the variable , which is obtained by introducing Lax pairs through the Lie algebra . 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  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  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  proposed another Lie algebra which is different from those in [9, 10] to deduce nonlinear integrable couplings. Wei and Xia  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  to deduce an integrable hierarchy (called the GC hierarchy) under the frame of zero curvature equations by the Tu scheme  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 . 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 . 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 then one gets It is well known that is a Lie algebra. A loop algebra of is given by where By using the loop algebra , introduce an isospectral problem : Set where The stationary equation admits that a solution for the is as follows: which gives rise to Set from (10) and (11) we have denote by We have The compatibility of the following Lax pair gives rise to where is a Hamiltonian operator.
By the trace identity presented in , we have Substituting the above results to the trace identity yields where Comparing the coefficients of of both sides in (20) gives It is easy to see . Thus, we have where are Hamiltonian conserved densities of the Lax integrable hierarchy (17). Therefore, we get a Hamiltonian form of the hierarchy (17) as follows: Let us consider the reduced cases of (17). When , we get that Taking , one gets a generalized Burgers equation:
3. The First Expanding Integrable Model of the GC Hierarchy
Zhang and Tam  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: where Define It is easy to compute that Set we have and are all simple Lie-subalgebras of the Lie algebra . The corresponding symmetric constant matrix appearing in the variational identity is that
A loop algebra corresponding to the Lie algebra is defined by We use the loop algebra to introduce a Lax pair:
The stationary equation is equivalent to from which we have Set we obtain from (37)
Note a direct calculation yields Therefore, zero curvature equation admits that Set , (44) reduces to the integrable hierarchy (17). When we take , we get an expanding nonlinear integrable model of the generalized Burgers equation (27) as follows: Obviously, the coupled equations are nonlinear with respect to the coupled variables and . 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 , , and their derivatives in the variable , where the functions , satisfy (27). In particular, we take a trivial solution of (27) to be ; then (45) reduces to the following equations: When we set , the above equations reduce to the well-known heat equation.
It is easy to see that the Lie algebra is isomorphic to the Lie algebra if equipped with a commutator as follows: Thus, under the Lie algebra , the Lax pair (36) can be written as In terms of (48) and (49) we obtain that Substituting the above results into the variational identity yields where Comparing the coefficients of on both sides in (51) gives From (37) we have . Thus, we get that where Therefore, we obtain the Hamiltonian structure of the nonlinear integrable coupling (44) as follows: where 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 It is easy to see that If we set , , and , then we have that Hence, the integrable couplings of the GC hierarchy cannot be generated by the Lie algebra 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 where , .
Solving the stationary zero curvature equation gives rise to Let , , , and ; then one gets from (62) that Noting , one infers that .
Equation (60) can be written as By computing that thus, we have
Substituting the above consequences into the trace identity proposed by Tu  yields that Comparing the coefficients of gives
Inserting the initial values in (62) gives . Therefore, we obtain that where are conserved densities of the expanding integrable model (65). Thus, (65) can be written as the Hamiltonian structure where , , is a Hamiltonian operator.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the Natural Science Foundation of China (11371361) and the Natural Science Foundation of Shandong Province (ZR2012AQ011, ZR2012AQ015).
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