^{1}

^{2,3}

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A lattice hierarchy with self-consistent sources is deduced starting from a three-by-three discrete matrix spectral problem. The Hamiltonian structures are constructed for the resulting hierarchy. Liouville integrability of the resulting equations is demonstrated. Moreover, infinitely many conservation laws of the resulting hierarchy are obtained.

Nonlinear integrable systems of the discrete version, treated as models of some physical phenomena, have attracted more and more attention in recent years. A well-known result is that a hierarchy of soliton equations can be generated through the isospectral compatibility condition of a pair of spectral problems [

With the development of soliton theory, people began to focus on the soliton equations with self-consistent sources. Soliton equations with self-consistent sources 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. Many integrable coupling systems with self-consistent sources in continuous cases are obtained [

In the present paper, first, a new three-by-three discrete matrix spectral problem is proposed. By means of constructing a proper continuous time evolution equation and using the discrete zero curvature equation a hierarchy of lattice models is derived. Then the Hamiltonian forms of the resulting hierarchy are worked out by using the discrete trace identity. Further, the Liouville integrability of the discrete systems is demonstrated. Infinitely many conservation laws and self-consistent sources for the integrable systems are also obtained.

We first recall some presentations on a discrete integrable system. For a lattice function

In this paper, we consider the

To get a hierarchy of lattice models associated with (

The initial values are taken as

So the discrete spectral problem (

Now we would like to derive the Hamiltonian structures for (

Set

By substituting

Therefore, we have

The lattice equations in (

We can get the following alternative form from (

In this section, we will construct the lattice hierarchy (

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the Project Global Change and Air-Sea Interaction (GASI-03-01-01-02), Nature Science Foundation of Shandong Province of China (no. ZR2013AQ017), Science and Technology Plan Project of Qingdao (no. 14-2-4-77-jch), Open Fund of the Key Laboratory of Ocean Circulation and Waves, Chinese Academy of Science (no. KLOCAW1401), and Open Fund of the Key Laboratory of Data Analysis and Application, State Oceanic Administration (no. LDAA-2013-04).