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The relation between the 3 × 3 complex spectral problem and the associated completely integrable system is generated. From the spectral problem, we derived the Lax pairs and the evolution equation hierarchy in which the coupled nonlinear Schr?dinger equation is included. Then, with the constraints between the potential function and the eigenvalue function, using the nonlineared Lax pairs, a finite-dimensional complex Hamiltonian system is obtained. Furthermore, the representation of the solution to the evolution equations is generated by the commutable flows of the finite-dimensional completely integrable system.

As is well known, the technique of the nonlinearization of Lax pairs has been a powerful tool for the finding of integrable systems in the last two decades or so. With this technique, the representation of the solution to the systems can also be generated. A lot of researches have been made in this way [

In this paper, we present a 3 × 3 AKNS matrix spectral problem

where the potential

spectral problem and the associated completely integrable system is considered. We derived the related evolution equation hierarchy, one of which is often referred to on the literature as the coupled nonlinear Schrödinger equation:

which is used by Manakov for studying the propagation of the electric field in a waveguide [

Now, suppose

where

The auxiliary problem of the spectral problem is set as follows:

Let

So the Lenard recursive sequence

is obtained and the Lenard recursive equation

is given, where K and J are two bi-Hamiltonian operators [

and

The isospectral evolution equations are

By (2.1)-(2.5), we have

Theorem 2.1.

is the Lax forms of evolution Equation (2.5). In other words, the hierarchy of solition Equation (2.5) is a isospectral compatible condition of (2.6).

Especially, if we take

By (2.5),

if

In order to give the constraints between the potential and the eigenfunction, First, the complex representation of the Poisson bracket is discussed.

The Poisson bracket of the real-valued function

The Poisson bracket of the complex-valued function F, H in the symplectic space

Lemma 3.1. [

then the symplectic form

Especially, if

are equivalent to the real Hamiltonian canonical equation

Which plays an important role in the generation of the completely integrable system in the Liouville sense.

Consider the spectral problem (1.1) and it’s adjoint spectral problem

A direct calculation shows that

where

and

Now, suppose

Set

Substituting (3.6) into (3.4), (3.5), we can get the

Hamiltonian function

where

If the coordinates are as follows:

The following theorem immediately holds.

Theorem 3.2. On the constraint (3.6), (3.4) and (3.5) with their conjugate representations are equal to the Hamiltonian canonical system

Define the Hamiltonian function as follows:

where

By (3.13)-(3.15), the following theorem hold.

Theorem 3.2. On the constraint (3.6), (3.4) and (3.5) with their conjugate representations are equal to the Hamiltonian canonical system

Theorem 3.4. [

satisfies the evolution Equation (2.5).

This project is supported by the Doctoral Scientific Research Foundation of Shijiazhuang University (No. 11BJ009).

Lanxin Chen,Junxian Zhang, (2015) A Finite-Dimensional Integrable System Related to the Complex 3 × 3 Spectral Problem and the Coupled Nonlinear Schrödinger Equation. World Journal of Engineering and Technology,03,322-327. doi: 10.4236/wjet.2015.33C048