Two New Integrable Hierarchies and Their Nonlinear Integrable Couplings

By introducing an invertible linear transform, a new Lie algebra G is obtained from the Lie algebra H. Making use of the compatibility conditions of the respective isospectral problems, a generalized NLS-MKdV hierarchy and a new integrable soliton hierarchy are achieved by using the trace identity or the variational identity. Then, two special non-semisimple Lie algebras H and G are explicitly conducted. As an application, the nonlinear continuous integrable couplings of the obtained integrable systems as well as their bi-Hamiltonian structures are established, respectively.


Introduction
Integrable equations are a significant research topic of classical integrable systems.Thereinto, integrable coupling, as an extension of the integrable equation, was formulated and initialized with the clarity of the inner relationship between Virasoro algebras and hereditary operators [1] [2].A few methods were presented by using perturbations [1] [2], enlarging spectral problems [3] [4] [5], creating higher-dimensional loop algebras [6] [7], constructing a new algebraic system [8] [9] [10], and making use of semi-direct sums of specific Lie algebras, for instance, the orthogonal Lie algebra ( ) 3, so R , to construct some soliton hierarchies and their integrable couplings [11]- [16].Much richer mathematical structures behind integrable couplings were explored, such as Lax pairs with several spectral parameters [17] [18] [19], integrable constrained flows with higher multiplicity [20] [21], local bi-Hamiltonian structures in higher dimen-sions and hereditary recursion operators of higher order [22] [23].A lot of complex physical phenomena can be explained by all kinds of coupling systems [24].
Therefore, integrable couplings have attracted more and more attention from researchers in engineering and mathematical theory.
Thereinto, the nonlinear integrable couplings are a charming subject, which can be achieved by using an extended Lie algebra.First, an isospectral problem , , , x s s i U U U u e u e u e u e e A φ φ φ and its auxiliary condition admit a zero curvature equation [ ] i.e., a Lax integrable system ( ), where ( ) , A  is the corresponding loop algebra of a Lie algebra A. Next, take enlarged spectral matrices and , 0 0 in which U and V derive from ( 1) and ( 2), respectively, where , , , p v v v v =  and u consist of u and v.
Then an enlarged zero curvature equation , , i.e., is a nonlinear integrable coupling of (3), because the commutator [ ] , c c U V can generate nonlinear terms.
In this paper, a new four-dimensional Lie algebra H is firstly presented, and another one G is obtained through an invertible linear transformation.A generalized NLS-MKdV hierarchy and a new integrable soliton hierarchy are achieved by using the Loop algebras H  and G  of H and G in Section 2. Two special non-semisimple Lie algebras H and G are determined programmat- ically, and its associated nonlinear continuous integrable couplings and their bi-Hamiltonian structures are established in Section 3. Finally, concluding remarks are given, as well as some proposals for the future work.

h h h h h h h g h h h h h h h h h h
An invertible linear transformation can be established as follows: ( ) , and , det 0, Specially, taking results in a new Lie algebra , where g g g g g g g g g g g g g g g g g In what follows, the corresponding loop algebras H  and G  of the Lie al- gebras H and G are introduced, respectively.Let , where a loop algebra ,

h m i h n j h m i h n j h h m i h n j h h m i h n j h h m i h n j h h m i h n j h
, .
Similarly, the commutator relations of the loop algebra G  have i j i j i j i j i j i j i j i j i j i j i j i j g m i g n j g m i g n j g g m i g n j g g m i g n j g g m i g n j g g m i g n j g Note that the commutator operations in loop algebras G  and W  are closed.
In the following section, one endeavor to deduce two soliton hierarchies by using the two Lie algebras.

Two New Integrable Hierarchies
the stationary zero curvature equation admits the recurrence relations for W as follows: ∑ Then (18) can be reset below: A direct calculation has , 0 , the zero curvature equation , , leads to the following integrable hierarchy where Taking 0 q = and the modified term 0 n ∆ = , the system (23) reduces to the NLS-MKTV hierarchy [25].Therefore, ( 23) is named a generalized NLS-MKdV hierarchy.

A New Integrable Hierarchy
( ) the stationary zero curvature [ ] admits the recurrence relations .
, the zero curvature equation , where

Bi-Hamiltonian Structures of (23) and (28)
In this section, the bi-Hamiltonian structures of soliton hierarchies ( 23) and (28) are established.Firstly, the bi-Hamiltonian structure of ( 23) is obtained by applying the trace identity. Letting can be obtained by the calculation of ( ) (A and B are square matrices).Substituting these results into the trace identity Comparing of the coefficients of 1 n λ − − these both sides of the above equa- tions, one has .
To fix the γ , taking 1 n = into (32) results in 0 γ = .Therefore, the bi-Hamiltonian structure of ( 23) can be obtained below: where, . 1 It is easy to prove * JL L J = .Therefore, the hierarchy (33) is integrable in the Liouville sense.
Next, the bi-Hamiltonian structure of ( 28) is derived by using the trace identity.Letting are computed, and Comparing the coefficients of 4 .
To fix the γ , substituting 1 n = into (35) results in 0 γ = .Therefore, the bi-Hamiltonian structure of (28) can be established as follows: ( ) , , , where, . 1 It is easy to prove Therefore, the hierarchy (36) is integrable in the Liouville sense.

Extension of Lie Algebras
, it is an extension of the Lie algebra H, where which is a critical factor for generating nonlinear integrable couplings of integrable hierarchies.In order to seek a nonlinear integrable coupling of ( 23), a loop algebra of the Lie algebra H reads , where Taking { } span , , , G g g g g = results in And corresponding loop algebra of G , which is enslaved to derive a integrable coupling of (28).

Nonlinear Integrable Coupling of the Generalized NLS-MKdV Hierarchy
An enlarged spectral matrix associated with the loop algebra H  is introduced as follows: i.e. ( , where U is defined by (16).Assume that

W a h m b h m c h m d h m e h m f h m
; )) .

V U V c h b h e h b h
, , leads to the following integrable hierarchy If 0 u v p = = = , the system (46) is reduced to (23).According to the concept of nonlinear integrable couplings [26] [27] [28], ( 46) is a nonlinear integrable coupling of (23).

Nonlinear Integrable Coupling of the Hierarchy (28)
An enlarged spectral matrix associated with the loop algebra g  is given below: where 1 U is defined by (25).Assume that ,   ( ) ( ) ( ) x x b r c s a c s q s a s r b r q r b r q r q r q r r s c s q s q s q s r s a r s s r q s r d    52) is reduced to (28), and (52) is a nonlinear integrable coupling of (28).

Bilinear Forms
In this section, the bi-Hamiltonian structures of the nonlinear integrable couplings of the generalized NLS-MKdV hierarchy (23) and the new integrable hierarchy ( 28) can be established.In order to achieve this target, two non-degenerate, symmetric and ad-invariant bilinear forms on two Lie algebras H and G are introduced.First of all, an isomorphic mapping σ between the Lie algebra H and a vector space R 8 is established that ( ) which imports a Lie algebraic system on R 8 .The corresponding commutator

Conclusion
The generalized NLS-MKdV hierarchy and its bi-Hamiltonian structure, reduced to the NLS-MKdV hierarchy [25], are derived from a new Lax pair.Based on the loop algebra of a new Lie algebra G, a spectral matrix is devised, and an integrable hierarchy and its bi-Hamiltonian structure are established; this is a new integrable system and not found in the related literature.Making use of extension forms of two Lie algebras, two nonlinear integrable couplings are achieved, and their Hamiltonian structures are constructed by using the Variational identity.Darboux transformations of the two integrable hierarchies can be embarked and constructed for exact solutions in the future.
. The first few sets are above-mentioned recursion relation uniquely engenders all differential polynomial functions , , , , m m m m m a b c d e and , 0 m f m ≥ .The first few sets are listed as follows:

H
. Chang, Y. X. Li DOI: 10.4236/jamp.2018.661131356 Journal of Applied Mathematics and Physics are ascertained in accordance with the symmetric property , , the Lie algebra G , where

1 U
is defined by (47).Substituting (72) into the Variational identity, and comparing the coefficients of to see 0 γ = , the adjoint symmetrical function of the integrable sys-

2.2.1. Generalized NLS-MKdV Hierarchy
) are non-degenerate, symmetric and ad-invariant associated with the Lie product.