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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 [

Recently, with the development of integrable systems, integrable couplings have attracted much attention. Integrable couplings [

Soliton equation with self-consistent sources (SESCS) [

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. [

This paper is organized as follows. In Section

Tu [

While we use Lie algebras to generate integrable hierarchies of evolution equations, we actually employ their loop algebras

We consider an auxiliary linear problem as follows:

The compatibility of (

Now, we consider Li soliton hierarchy [

To establish the nonlinear integrable coupling system of the Li soliton hierarchy, the adjoint equation

Taking

So, we can say that the system in (

According to (

Based on the result in [

For

According to (

When

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

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).

^{4}equation using lie symmetry approach along with the simplest equation and Exp-function methods