Integrable Coupling of Expanded Isospectral and Non-Isospectral Dirac Hierarchy and Its Reduction

: In this paper, we ﬁrst generalize the Dirac spectral problem to isospectral and non-isospectral problems and use the Tu scheme to derive the hierarchy of some new soliton evolution equations. Then, integrable coupling is obtained by solving the isospectral and non-isospectral zero curvature equations.We ﬁnd that the obtained hierarchy has the bi-Hamiltonian structure of the combined form. In particular, one of the integrable soliton hierarchies is reduced to be similar to the coupled nonlinear Sch¨ o rdinger system in the AKNS hierarchy. Next, the strict self-adjointness of the reduced equation system is veriﬁed, and conservation laws are constructed with the aid of the Ibragimov method. In addition, we apply the extended Kudryashov method to obtain some exact solutions of this reduced equation system.


Introduction
Integrable systems are an important research field in nonlinear science and have important application backgrounds in the fields of optical fiber communication, superconductivity, shallow water waves, and plasma. The famous Lax pair method proposed by Magri [1] can generate a large number of integrable hierarchies of evolutionary equations. Tu [2] constructed the Lax pair or isospectral problem by using the Lie algebra of square matrix and its corresponding loop algebra, derived the soliton hierarchy from its compatibility, and determined the Hamiltonian structure of the soliton hierarchy by the trace identity, which is called the Tu scheme by Ma [3]. Many interesting isospectral integrable hierarchies and their properties can be obtained using the Tu scheme, as shown in Refs. [4][5][6][7][8][9][10][11][12][13]. The above integrable hierarchies are proposed in the case of isospectral problems. As a result, Zhang et al. [14][15][16] proposed a method for generating a nonisospectral integrable hierarchy on the basis of the assumption that λ t = ∑ m i=0 k i (t)λ m−i . With the introduction of the integrable coupling problem, there have been more methods to construct integrable coupling, such as the perturbation method, the generalization method of new loop algebra, and the non-semisimple Lie algebra method. Among them, Ma et al. [17], for the first time, put forward the use of the non-semisimple Lie algebra method to find integrable coupling. Based on this, experts and scholars worldwide have obtained meaningful conclusions, which have greatly promoted the development of integrable coupling [18][19][20][21][22].
In this paper, we first consider the application of the Tu scheme to the Dirac spectral problem and derive some new isospectral-non-isospectral soliton equation hierarchies. In Section 3, we construct a new Lie algebra g and its corresponding loop algebra g and obtain the nonisospectral integrable coupling of Dirac by solving the zero curvature equation.
In Section 4, we use the method proposed by Tu [2] to generate integrable Hamiltonian hierarchies by using trace identities, as well as the Tu scheme to obtain some properties of in-tegrable systems and Hamiltonian structures, such as the works [23][24][25]. In Sections 5 and 6, we focus on Equation (12) that is reduced from the new isospectral-non-isospectral soliton equation hierarchies in Section 2, which are similar to the coupled nonlinear Schördinger system in the AKNS hierarchy; then, we study the self-adjointness and conservation laws of this equation system using the method proposed by Ibragimov [26] and find some exact solutions to the equation system by improving the methods in Refs. [27][28][29].

An Isospectral-Non-Isopectral Dirac Equation Integrable Hierarchy
Firstly, we show a classical algebra A 1 = h, e, f , where given a loop algebra Consider the following Dirac spectral problem where Taking the spectral evolution λ t = ∑ j≥0 k j (t)λ −j , the compatibility condition of (1) and According to the generalized Tu scheme, we first solve the stationary zero curvature equation for V: which gives rise to Note that then, a direct calculation reads ; then, by the compatibility condition of non-isospectral Lax pairs, we have which admits an isospectral-non-isospectral integerable hierarchy of evolution equations From (5) we find a n c n =: L a n c n .
Based on the loop algebrag, we consider the spectral problem where u = (p, q, r, s) T is the potential, In terms of the steps of the Tu scheme, the non-isospectral stationsry zero curvature equation which has an equivalent form Set then, the first two sets are written as: Then, Equation (15) By taking (18) into (19) and taking these terms of gradation 0, we can obtain −W [m] Then, the zero curvature equation gives rise to the non-isospectral Dirac integrable coupling hierarchy as follows: The first two nonlinear examples as: and

Hamiltonian Structure of the Dirac Integrable Coupling
In this section, we focus on the Hamiltonian structure of the hierarchy (22) by using the trace identity proposed by Tu [2]. We denote the trace of the square matrices A and B by < A, B >= tr(AB). From (14), let we obtain which can be substituted into the following two sets of component-trace identity Then, we substitute the Laurent series into the above identity (25), (26), and we compare the powers of λ to yield We find that γ = 0 via substituting m = 1 into (27). Hence, Consequently, we obtain the Hamilitonian structure of (22) with the Hamiltonian operators and the Hamiltonian functionals For the first component, one has the Hamiltonian structure for the integrable coupling hierarchy (22) as follows: Now, using the recurrence relationships (17), we obtain the recursion operator N of the hierarchy (22) as follows: where Therefore, we obtain the first component of the Hamiltonian structure of the soliton hierarchy (22) as follows: where the second pair of Hamiltonian operators and the recursion operator Simliarily, we can obtain the following relationships and for the second component, one has the Hamiltonian structure for the integrable coupling hierarchy (22) as follows: where the second pair of Hammiltonian operators and the recursion operator After calculation, we find

Self-Adjointness and Conservation Laws
In this section, we consider the strictly self-adjointness and conservation laws of Equation (12) by the method proposed by Ibragimov [26].
First, let us start with the following notation and basic definition where x k (k = 1, 2, · · · , n) are independent variables, and D i denotes the total differentiations operator with four dependent variables, u, v, u, and v, i.e., The systems of m differential equations can be written as F α (x, u, u (1) , . . . , u (s) ) = 0, α = 1, · · · , m, which admits the adjoint equations where with L as the formal Lagrangian for Equation (37) given by and δ δu α is the variational derivative that reads It means that the equation holds with a certain (in general,variable) coefficient λ. (12) is strictly self-adjoint.

Proof. Set
then, the formal Lagrangian for (40) can be written as which admits the adjoint Substituting Thus, Equation (42) is strictly self-adjoint. Next, we consider the conservation laws of Equation (12). Let X = ξ i ∂ ∂x i + η α ∂ ∂u α be any Lie-Bäcklund operator where (47) We associate with X the following n operators N i (i = 1, . . . , n) by the formal sums: where The Euler-Lagrange (41), Lie-Bäcklund (46), and the associated operators (49) are connected by the following fundamental identity (see [31]): . . , u (s) ) ∈ A with several independent variables x = (x 1 , . . . , x n ) and several dependent variables u = (u 1 , . . . , u m ) is the divergence of a vector field H = (h 1 , . . . , h n ), h i ∈ A, where A represents the set of all finite order differential functions, i.e., of the Frobenius Equation (37) leads to the conservation law D i (C i ) = 0 constructed by the formula with W α = η α − ξ j u α j , α = 1, . . . , m.

Proof. Let us begin with the Euler-Lagrange equations
where L(x, u, u (1) ) is a first-order Lagrangian, i.e., it involves, along with the independent variables x = (x 1 , . . . , x n ) and the dependent variables u = (u, . . . , u m ), the first-order derivatives u (1) = {u α i } only.
Noether's theorem states that if the variational integral with Lagrangian L(x, u, u (1) ) is invariant under a group G with a generator then the vector field C = (C 1 , . . . , C n ), defined by provides a conservation law for the Euler-Lagrange Equation (54), i.e., obeys the equation div C ≡ D i (C i ) = 0 for all solutions of (54), i.e., Any vector field C i , satisfying (57), is called a conserved vector for Equation (54).
In order to apply Noether's theorem, one has first of all to find the symmetries of Equation (54). Then, one should single out the symmetries leaving invariant the variational integral (54). This can be done by means of the following infinitesimal test for the invariance of the variational integral (see [31]): where the generator X is prolonged to the first derivatives u (1) by the formula If Equation (58) is satisfied, then the vector (56) provides a conservation law. From Lemma 1, we can obtain that if one adds to a Lagrangian the divergence of any vector field, the Euler-Lagrange equations remain invariant. Therefore, one can add to the Lagrangian L the divergence of an arbitrary vector field depending on the group parameter and replace the invariance condition (58) by the divergence condition Then, Equation (54) is again invariant and has a conservation law D i (C i ) = 0, where (56) is replaced by It follows from Equations (50) and (58) that if a variational integral L dx with a higherorder Lagrangian L(x, u, u (1) , u (2) , . . .) is invariant under a group with a generator (59), then the vector provides a conservation law for the corresponding Euler-Lagrange equations. Dropping the differentiations of L with respect to higher-order derivative u (4) , . . . and changing the summation indices, we obtain from (62) and (48): where N i is the operator (48), and W α = η α − ξ j u α j is given by (49).
With the aid of Maple, Equation (42) has five symmetries as follows: For the generator X 3 , we have W 1 = v, and W 2 = −u.Thus, formulas (53) yield the following conserved vector For the generator X 1 , we have W 1 = −u x , and W 2 = −v x . Thus, formulas (53) yield the following conserved vector For the generator X 2 , we have W 1 = −u t , and W 2 = −v t . Thus, formulas (53) yield the following conserved vector For the generator X 4 , we have W 1 = −vx − tu x , and W 2 = xu − tv x . Thus, formulas (53) yield the following conserved vector For the generator X 5 , we have Thus, formulas (53) yield the following conserved vector

Exact Solutions of Equation (12)
Let us take a nonlinear partial differential equation (NLPDE) of the form F(x, u t , u x , u tt , u xx , . . .) = 0.

Conclusions and Discussion
In this paper, the Lax pairs of Dirac spectral problems were studied by the improved Tu-scheme of Zhang et al., and the isospectral-non-isospectral integrable hierarchies were derived from their compatibility. The integrable hierarchy can be reduced to Equation (12). Subsequently, the self-adjointness and conservation laws of this system of equations were discussed using the method proposed by Ibragimov, and the exact solution of the equation system was obtained by using the solutions of the Bernoulli and Riccati equations. In Sections 3 and 4, based on the non-semisimple Lie algebrag, we obtained the nonisospectral coupling hierarchies of the soliton hierarchy (22) and the Hamiltonian structure of its coupling hierarchy. We hope to apply the Tu scheme to find new nonisospectral integrable systems and their properties.