Self-Consistent Sources and Conservation Laws for Nonlinear Integrable Couplings of the Li Soliton Hierarchy
New explicit Lie algebras are introduced for which the nonlinear integrable couplings of the Li soliton hierarchy are obtained. Then, the nonlinear integrable couplings of Li soliton hierarchy with self-consistent sources are established. Finally, we present the infinitely many conservation laws for the nonlinear integrable coupling of Li soliton hierarchy.
Soliton theory has achieved great success during the last decades, it is being applied to many fields. The diversity and complexity of soliton theory enables investigators to do research from different views, such as binary nonlinearization of soliton hierarchy  and Bäcklund transformations of soliton systems from symmetry constraints .
Recently, with the development of integrable systems, integrable couplings have attracted much attention. Integrable couplings [3, 4] are coupled systems of integrable equations, which have been introduced when we study of Virasoro symmetric algebras. It is an important topic to look for integrable couplings because integrable couplings have much richer mathematical structures and better physical meanings. In recent years, many methods of searching for integrable couplings have been developed [5–13], but all the integrable couplings obtained are linear for the . As for how to generate nonlinear integrable couplings, Ma proposed a general scheme . Suppose that an integrable system has a Lax pair and , which belong to semisimple matrix Lie algebras. Introduce an enlarged spectral matrix from a zero curvature representation where then we can give rise to This is an integrable coupling of (1), and it is a nonlinear integrable coupling because the commutator can generate nonlinear terms.
Soliton equation with self-consistent sources (SESCS) [15–22] is an important part in soliton theory. Physically, the sources may result in solitary waves with a nonconstant velocity and therefore lead to a variety of dynamics of physical models. For applications, these kinds of systems are usually used to describe interactions between different solitary waves and are relevant to some problems of hydrodynamics, solid state physics, plasma physics, and so forth. How to obtain an integrable coupling of the SESCS is an interesting topic; in this paper, we will use new formula  presented by us to generalize soliton hierarchy with self-consistent sources.
The conservation laws play an important role in discussing the integrability for soliton hierarchy. An infinite number of conservation laws for KdV equation was first discovered by Miura et al. . The direct construction method of multipliers for the conservation laws was presented , the Lagrangian approach for evolution equations was considered in , Wang and Xia established the infinitely many conservation laws for the integrable super hierarchy , and the infinite conservation laws of the generalized quasilinear hyperbolic equations were derived in . Comparatively, the less nonlinear integrable couplings of the soliton equations have been considered for their conservation laws.
This paper is organized as follows. In Section 2, a kind of explicit Lie algebras with the forms of blocks is introduced to generate nonlinear integrable couplings of Li soliton hierarchy. In Section 3, a new nonlinear integrable coupling of Li soliton hierarchy with self-consistent sources is derived. In Section 4, we obtain the conservation laws for the nonlinear integrable couplings of Li hierarchy. Finally, some conclusions are given.
2. Lie Algebras for Constructing Nonlinear Integrable Couplings of Li Soliton Hierarchy
Tu  presented a base of the Li algebra as follows: where which have the commutative relations Let us introduce a Lie algebra with matrix blocks by using in order to get nonlinear couplings of soliton hierarchy as follows: where Define a commutator as follows: A direct verification exhibits that Set then we find that and and are all simple Lie subalgebras.
While we use Lie algebras to generate integrable hierarchies of evolution equations, we actually employ their loop algebras to establish Lax pairs, where represents a set of Laurent ploynomials in and is a Lie algebra. Based on this, we give the loop algebras of (9) as follows: where , , , , .
We consider an auxiliary linear problem as follows: where , is a pseudoregular element, , and are field variables defined on , , .
The compatibility of (16) gives rise to the well-known zero curvature equation The general scheme of searching for the consistent , and generating a hierarchy of zero curvature equations was proposed in . Solving the following equation: then we sesrch for , the new can be constructed by Solving zero curvature (17), we could get evolution equation as follows:
Now, we consider Li soliton hierarchy . In order to set up nonlinear integrable couplings of the Li soliton hierarchy with self-consistent sources, we first consider the following matrix spectral problem: that is, where is a spectral parameter and satisfies which is matrix spectral problem of the Li soliton hierarchy .
To establish the nonlinear integrable coupling system of the Li soliton hierarchy, the adjoint equation of the spectral problem (21) is firstly solved, we assume that a solution is given by the following: Therefore, the condition (18) becomes the following recursion relation: Choose the initial data we see that all sets of functions , and are uniquely determined. In particular, the first few sets are as follows: Considering From the zero curvature equation , we obtain the nonlinear integrable coupling system with the Hamiltonian operator and the hereditary recursion operator , respectively, as follows: where Obviously, when in (28), the above results become Li soliton hierarchy. So, we can say that (28) is integrable coupling of the Li soliton hierarchy.
Taking , we get that the nonlinear integrable couplings of Li soliton hierarchy are as follows:
So, we can say that the system in (28) with provides a hierarchy of nonlinear integrable couplings for the Li hierarchy of the soliton equation.
3. Self-Consistent Sources for the Nonlinear Integrable Couplings of Li Soliton Hierarchy
For we define that where and is a constant.
According to (34) and (36), we obtain a kind of nonlinear integrable couplings with self-consistent sources as follows: Therefore, according to formulas (34) and (36), we have the following results by direct computations: by taking and in formulas (34) and (36). Therefore, we have nonlinear integrable coupling system of the Li equations hierarchy with self-consistent sources as follows: with where , , and is the standard inner product in .
When and, we obtain nonlinear integrable couplings of Li hierarchy with self-consistent sources with
4. Conservation Laws for the Nonlinear Integrable Couplings of Li Soliton Hierarchy
In what follows, we will construct conservation laws for the nonlinear integrable couplings of the Li hierarchy. For the coupled spectral problem of Li hierarchy we introduce the variables From (44), we have We expand , , and in powers of as follows: Substituting (47) into (46) and comparing the coefficients of the same power of , we obtain the following: and a recursion formula for , , and as follows: Because of where Assume that Then, (50) can be written as and , which are the right form of conservation laws. We expand , and as series in powers of with the coefficients, which are called conserved densities and currents, respectively: where and are constants of integration. The first conserved densities and currents are read as follows: The recursion relations for , and are as follows: where , , and can be calculated from (49). The infinite conservation laws of nonlinear integrable couplings (37) can be easily obtained in (45)–(55), respectively.
In this paper, a new explicit Lie algebra was introduced, and a new nonlinear integrable couplings of Li soliton hierarchy with self-consistent sources was worked out. Then, the conservation laws of Li soliton hierarchy were also obtained. The method can be used to other soliton hierarchy with self-consistent sources. In the near future, we will investigate exact solutions of nonlinear integrable couplings of soliton equations with self-consistent sources which are derived by using our method.
The study is supported by the National Natural Science Foundation of China (Grant nos. 11271008, 61072147, and 1071159 ), the Shanghai Leading Academic Discipline Project (Grant no. J50101), and the Shanghai University Leading Academic Discipline Project (A. 13-0101-12-004).
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