﻿ 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
Vol.03 No.03(2015), Article ID:60554,6 pages
10.4236/wjet.2015.33C048

A Finite-Dimensional Integrable System Related to the Complex 3 × 3 Spectral Problem and the Coupled Nonlinear Schrödinger Equation

Lanxin Chen, Junxian Zhang

Shijiazhuang University, Shijiazhuang 050035, China

Email: LXCHENmath@163.com  Received 6 August 2015; accepted 15 October 2015; published 22 October 2015 ABSTRACT

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 Schrödinger 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.

Keywords:

Integrable System, Evolution Equation, Spectral Problem, Commutable Flows 1. Introduction

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 -. So far higher order matrix spectral problem and complex spectral problem are all attractive to the mathematical and physical science. However, due to theoretical difficulty and the complexity of computation, the relevant research is relatively rare for the present.

In this paper, we present a 3 × 3 AKNS matrix spectral problem (1.1)

where the potential , are complex-valued potential functions, is a complex spectral parameter, . The relation between this 3rd-order complex

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: (1.2)

which is used by Manakov for studying the propagation of the electric field in a waveguide . Each equation governs the evolution of one of the components of the field transverse to the direction of propagation. Also it can be derived as a model for wave propagation under conditions similar to those where nonlinear Schrödinger equation applies and there are two wavetrains moving with nearly the same group velocity . In recent years, this system is widely studied   and used as a key model in the field of optical solitons in fibers   to explain how the solitons waves transmit in optical fiber, what happens when the interaction among optical solitons influences directly the capacity and quality of communication and so on -.

2. The Evolution Equations and Their Lax Pairs

Now, suppose is the basic interval, if and their derivatives on x are all decay at infinity, then ; if they are all periodic functions, then . In order to get the evolution equations, we first solve the stationary zero-curvature equation: (2.1)

where (2.2)

The auxiliary problem of the spectral problem is set as follows: (2.3)

Let and the initial value (2.4)

So the Lenard recursive sequence is obtained and the Lenard recursive equation

(2.5)

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

and

The isospectral evolution equations are

(2.6)

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

(2.7)

(2.8)

(2.9)

By (2.5),

if, it is exactly the coupled nonlinear Schrödinger Equation (1.2) which is a well-known equation and is of great value in physics (where the symbol * denotes the complex conjugate).

3. A Finite-Dimensional Hamiltonian System

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 in the symplectic space is defined as follows:

The Poisson bracket of the complex-valued function F, H in the symplectic space is defined as follows:

Lemma 3.1.  Let

then the symplectic form can be written the complex Poisson bracket is equivalent to the real Poisson bracket when the Hamiltonian functions H and F are all real-valued functions, namely,

Especially, if, the complex Hamiltonian canonical equation

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

(3.1)

A direct calculation shows that

where. Then 

(3.2)

and

(3.3)

Now, suppose is an eigenvalue of (1.1) and (3.1), are the eigenfunctions for,. Then, the spectral problem (1.1) and it’s adjoint spectral problem (3.1) can be rewritten as follows:

(3.4)

(3.5)

Set. We consider the following constraint:

(3.6)

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

Hamiltonian function

(3.7)

where

If the coordinates are as follows:

(3.8)

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

(3.9)

Define the Hamiltonian function as follows:

(3.10)

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

(3.11)

Theorem 3.4.   Suppose is an involutive solution of the Hamiltonian canonical equation systems (3.9) (3.11), then

satisfies the evolution Equation (2.5).

Acknowledgements

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

Cite this paper

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

References

1. 1. Cao, C.W. and Geng, X.G. (1990) Nonlinear Physics, Research Reports in Physics, Spring-Verlag, Berlin, 68-78. http://dx.doi.org/10.1007/978-3-642-84148-4_9

2. 2. Cao, C.W. (1990) Science in China, 33, 528.

3. 3. Cao, C.W. and Geng, X.G. (1990) J. Phys. A, 21, 4117.

4. 4. Manakov, S.V. (1974) JETP Sov.Phys., 38, 248.

5. 5. Yang, J. (1998) Multiple Permanent-wave Trains in Nonlinear Systems. Studies in Applied Mathematics, 100, 127-152. http://dx.doi.org/10.1111/1467-9590.00073

6. 6. Geng, X.G., Zhu, J.Y. and Liu, H. (2015) Initial-Boundary Value Problems for the Coupled Nonlinear Schr?dinger Equation on the Half-Line. Studies in Applied Mathematics, 135, 310-346. http://dx.doi.org/10.1111/sapm.12088

7. 7. Sun, Y., Sun, W.R., Xie, X.Y. and Tian, B. (2015) Bright Solitons for the (2+12+1)-Dimensional Coupled Nonlinear Schr?dinger Equations in a Graded-Index Waveguide. Communications in Nonlinear Science and Numerical Simulation, 29, 300-306. http://dx.doi.org/10.1016/j.cnsns.2015.05.009

8. 8. Evangelidis, S.G., Mollenauer, L.F., Gordon, J.P. and Bergano, N.S. (1992) Polarization Multiplexing with Solitons. Journal of Lightwave Technology, 10, 28-35. http://dx.doi.org/10.1109/50.108732

9. 9. Marcuse, D., Menyuk, C.R. and Wai, P.K.A. (1997) Application of the Ma-nakov-PMD Equation to Studies of Signal Propagation in Optical Fibers with Randomly Varying Birefringence. Jour-nal of Lightwave Technology, 15, 1735-1746. http://dx.doi.org/10.1109/50.622902

10. 10. Radhakrishnan, R. and Lakshmanan, M. (1995) Bright and Dark Soliton Solutions to Coupled Nonlinear Schrodinger Equations. Journal of Physics A: Mathematical and General, 28, 2683. http://dx.doi.org/10.1088/0305-4470/28/9/025

11. 11. Drazin, P.G. and Johnson, R.S. (1992) Solitons: An Introduction. Cambridge University Press, Cambridge.

12. 12. Radhakrishnan, R., Lakshmanan, M. and Hietarinta, J. (1997) Inelastic Collision and Switching of Coupled Bright Solitons in Optical Fibers. Physical Review E, 56, 2213. http://dx.doi.org/10.1103/PhysRevE.56.2213

13. 13. Peter, J.O. (1999) Applications of Lie Groups to Differential Equ-ations. 2nd Edition, Spring-Verlag New York Berlin Heidelberg.

14. 14. Adler, M. (1979) On a Trace Functional for Formal Pseudo-Differential Operators and the Symplectic Structure of the Korteweg-Devries Type Equations. Inven-tiones Mathematical, 50, 219-248. http://dx.doi.org/10.1007/BF01410079

15. 15. Gu, Z.Q. (1991) Complex Confocal Involutive Systems Associated with the Solutions of the AKNS Evolution Equations. Journal of Mathematical Physics, 32, 1498. http://dx.doi.org/10.1063/1.529256

16. 16. Cao, C.W., Geng, X.G. and Wu, Y.T. (1999) From the Special 2 + 1 Toda Lattice to the Kadomtsev-Petviashvili Equation. Journal of Physics A: Mathematical and General, 32, 8059. http://dx.doi.org/10.1088/0305-4470/32/46/306

17. 17. Cao, C.W. (1991) Acta. Math. Sinica, 7, 216.