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 WJET  Vol.3 No.3 C , October 2015
A Finite-Dimensional Integrable System Related to the Complex 3 × 3 Spectral Problem and the Coupled Nonlinear Schrödinger Equation
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 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.
Cite this paper: Chen, L. and Zhang, J. (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, 3, 322-327. doi: 10.4236/wjet.2015.33C048.
References

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

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

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

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

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

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

 
 
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