#### Abstract

Based on a well-known Lie algebra, the multicomponent Guo hierarchy with self-consistent sources is proposed. With the help of a set of non-semisimple Lie algebra, the nonlinear bi-integrable couplings of the multicomponent Guo hierarchy with self-consistent sources are obtained. It enriches the content of the integrable couplings of hierarchies with self-consistent sources. Finally, the Hamiltonian structures are worked out by employing the variational identity.

#### 1. Introduction

Since the notion of integrable couplings was proposed in view of Virasoro symmetric algebras [1, 2] and the soliton theory, considerable research has been reported on the integrable couplings and associated properties of some known interesting integrable hierarchies, such as the AKNS hierarchy, the KN hierarchy, and the Burger hierarchy [3–6]. In order to produce multicomponent integrable systems, a type of multicomponent loop algebra was structured, and it followed that some multicomponent integrable systems were given and Hamiltonian structures of multicomponent systems were constructed through the component-trace identity in [7–13] and a general structure between Lie algebras and integrable coupling was given in [14, 15]. Here it is necessary to point out that the above-mentioned integrable couplings are linear for the supplementary variable, so they are called linear integrable couplings. Recently, Ma proposed the notion of nonlinear integrable couplings and gave the general scheme to construct the nonlinear integrable couplings of hierarchies [16]. In [17, 18], based on the general scheme of constructing nonlinear integrable couplings, Zhang introduced some new explicit Lie algebras and obtained the nonlinear integrable couplings of the GJ hierarchy, the Yang hierarchy, and the CBB hierarchy. In [19], Ma employed a class of non-semisimple matrix loop algebras to generate bi-integrable couplings of soliton equation from the zero curvature equations.

On the other hand, with the development of soliton theory, people began to focus on the soliton equation with self-consistent sources [20, 21]. The soliton equations with self-consistent sources have important physical meaning and are often used to express interactions between different solitary waves and are relevant to some problems of hydrodynamics, solid state physics, plasma physics, and so on [22, 23]. In [24–26], Yu and Xia discussed the integrable couplings of hierarchies with self-consistent sources and obtained a series of integrable hierarchies with self-consistent sources and their couplings systems as well as Hamiltonian structures. In these papers, the integrable couplings are linear for supplementary variables.

As mentioned above, we will find that there are very few researches on the nonlinear integrable couplings of hierarchies with self-consistent sources available up till now, especially on the nonlinear bi-integrable couplings of multicomponent hierarchy. In our work, we will derive the nonlinear bi-integrable couplings of multicomponent Guo hierarchy with self-consistent sources and their Hamiltonian structures. It will enrich the content of the integrable couplings of hierarchies with self-consistent sources. The plan of the paper is as follows. In Section 2, the multicomponent Guo hierarchy with self-consistent sources is given. By employing the variational identity, the Hamiltonian structure of the above hierarchy is obtained. The notion of bi-integrable couplings is introduced and the nonlinear bi-integrable couplings of the multicomponent Guo hierarchy with self-consistent sources and their Hamiltonian structures are derived in Section 3. Finally, some conclusions are given in Section 4.

#### 2. Multicomponent Guo Hierarchy with Self-Consistent Sources

In this section, we recall briefly some basic notions from the theory of the multicomponent integrable system. If denotes a class of matrices where is the th column of the matrix , stands for an arbitrary positive integer, then is a linear space.

*Definition 2.1 (see [7]). * Let and , their product is defined as

*Definition 2.2 (see [7]). *Set and , a commutation operation is defined as
It is easy to verify that is a Lie algebra. Its corresponding loop algebra is
equipped with a commutation operation as follows:
Consider the following isospectral problem:
where are potentials, is the spectral parameter, , and . By means of constructing a proper time evolution
where . Then solving stationary zero curvature equation , one arrives at the recursion relation as follows:
For an arbitrary natural number , denoting
then the stationary zero curvature equation can be written as
a direct calculation reads
Then the zero curvature equation
gives the integrable system
where is a Hamiltonian operator, and the recurrence operator is given from (2.7) by
Accordingly, when , the system (2.12) reduces to the Guo hierarchy in [5]; when , the system (2.12) is a multicomponent Guo hierarchy. We observe that , the Hamiltonian functions are involutive with each other, and each is a common conserved density of the expression (2.12). Therefore, the multicomponent Guo hierarchy (2.12) is Liouville integrable.

In order to deduce the Hamiltonian structure of (2.12), we rewrite (2.2) as follows: where Solving the matrix equation for the constant matrix : , , presents Then in terms of , define a linear functional: It is easy to find that satisfies the variational identity By using the above linear functional (2.17), according the variational identity, we have Comparing the coefficients of yields Taking in the above equation gives .

Hence, the Guo hierarchy (2.12) can be written as a Hamiltonian form where .

Next, we will derive the Guo hierarchy with self-consistent sources. Consider the linear problem The following equation was given in [21] where are constants, and determines a finite-dimensional invariant set for the flow (2.21). Here we get [27] where , , , stands for the inner product in .

Then, according to (2.23) and taking , the Guo hierarchy with self-consistent sources can be rewritten as follows: Taking , we obtain the following multicomponent equations with self-consistent sources

#### 3. Nonlinear Bi-Integrable Couplings of Multicomponent Guo Hierarchy with Self-Consistent Sources

The multicomponent Guo hierarchy has been given in the above. In the section, we will construct the nonlinear bi-integrable couplings of multicomponent Guo hierarchy with self-consistent sources by following the way in [19], which is introduced as follows.

Given an integrable equation which possesses a zero curvature representation where , and taking , , then the enlarged zero-curvature equation equivalently yields We call (3.4) bi-integrable couplings of (3.1). Note that it is nonlinear with respect to the two supplementary variables , thereby providing candidates for nonlinear bi-integrable couplings.

Next, we will construct an expanding Lie algebra of the Lie algebra and its corresponding loop algebra. Consider a vector space , where is the th column of the matrix .

*Definition 3.1. *Set as two elements of , their commutation operation is defined as
It is easy to verify that consists of a Lie algebra along with (3.5). Its corresponding loop algebra is given by
equipped with a commutation operation as follows:
Denoting and , then we have
Here we note that the subalgebras and are both non-semisimple, which is very important for deriving the nonlinear integrable couplings of hierarchy.

In terms of the loop algebra , consider an isospectral problem as follows:
Set
where . A solution to gives
Similar to (2.9), we obtain
Therefore, the zero-curvature equation
admits the following bigger integrable system:
where is the Hamiltonian operator. Similar to (2.13), by employing (3.11), after tedious calculation, the recurrence operator can be obtained and here we omit it.

Taking in (3.14) reduces to an integrable coupling of the Guo hierarchy; taking in (3.14) reduces to another integrable coupling of the Guo hierarchy. Here we note that except the first variables , all integrable couplings presented above are nonlinear, since the supplementary systems (3.14) with are nonlinear with respect to the four dependent variables . So, in term of the definition of integrable couplings, we conclude that the system (3.14) is nonlinear bi-integrable couplings of the multicomponent Guo hierarchy.

In order to deduce to the Hamiltonian structure of (3.14), we rewrite (3.5) as follows: whereSolving the matrix equation for the constant matrix : , presents Then in terms of , define a linear functional It is easy to find that satisfies the variational identity (2.18). By using the above linear functional, we have Substituting (3.19) into the variational identity (2.18), we have Comparing the coefficients of yields Taking in the above equation gives .

Hence, the nonlinear bi-integrable couplings of the multicomponent Guo hierarchy (3.14) can be written as a Hamiltonian form where .

Next, we will derive the nonlinear bi-integrable couplings of the multicomponent Guo hierarchy with self-consistent sources. Consider the linear problem According to (2.24) and the definition of , we get where , , , ; ; stands for the inner product in .

Then, according to (2.23) and taking , the nonlinear bi-integrable couplings of the multicomponent Guo hierarchy with self-consistent sources can be rewritten as follows:

#### 4. Conclusions

In this paper, we derive the multicomponent Guo hierarchy with self-consistent sources and its nonlinear bi-integrable couplings by using the Lie algebra . As far as we know, the existing papers are not designed for constructing the nonlinear bi-integrable couplings of multicomponent hierarchy with self-consistent sources. This method can be used to derive the other multicomponent soliton hierarchies with self-consistent sources and their nonlinear bi-integrable couplings.

#### Acknowledgments

This work was supported by Innovation Project of Chinese Academy of Sciences (no. KZCX2-EW-209) and National Natural Science Foundation of China (no. 11271007).