Two Expanding Integrable Models of the Geng-Cao Hierarchy

and Applied Analysis 3


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][2][3][4][5][6][7][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   = (, V) 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  1 and  2 , which meets ( If the subalgebra  2 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 [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.

The GC Integrable Hierarchy and Its Hamiltonian Structure
We have known that then one gets It is well known that span{ℎ, , } =  1 is a Lie algebra.A loop algebra of  1 is given by where By using the loop algebra Ã1 , introduce an isospectral problem [13]: 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 [14], 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  = −1.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  = 1, we get that Taking  = 2, one gets a generalized Burgers equation: Remark 1.The Hamiltonian structure ( 23) is different from that in [14].We call (17) the GC hierarchy.

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: where Define It is easy to compute that Set we have 1 and  2 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 F 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  1 =  2 = 0, (44) reduces to the integrable hierarchy (17).When we take  = 2, 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  1 and  2 .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 , V, and their derivatives in the variable , where the functions , V satisfy (27).In particular, we take a trivial solution of (27) to be  = V = 0; then (45) reduces to the following equations: When we set  2 = 0, the above equations reduce to the wellknown heat equation.In order to deduce Hamiltonian structure of the nonlinear integrable coupling (44), we define a linear functional [11]: where  = ( 1 , . . .,  6 )  ,  = ( 1 , . . .,  6 )  .It is easy to see that the Lie algebra  is isomorphic to the Lie algebra  6 if equipped with a commutator as follows: Thus, under the Lie algebra  6 , 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  = −1.Thus, we get that where Therefore, we obtain the Hamiltonian structure of the nonlinear integrable coupling (44) as follows: where  is obviously Hamiltonian.

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  = span{ℎ 1 , . . ., ℎ 6 },  1 = span{ℎ 1 , ℎ 2 , ℎ 3 }, and  2 = span{ℎ 4 , ℎ 5 , ℎ 6 }, 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 ℎ  () = ℎ    ,  = 1, 2, 3, 4, 5, 6.