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.
Highlights
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
We present a 3 × 3 AKNS matrix spectral problem iξ u1 u2
Each equation governs the evolution of one of the components of the field transverse to the direction of propagation. 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 [5]. This system is widely studied [6] [7] and used as a key model in the field of optical solitons in fibers [8] [9] 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 [10]-[12]
Summary
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. Zhang where the potential u = (u1, v1,u2 , = v2 )T , u1 u1= (x,t),u2 u= 2 (x,t), v1 v= 1(x,t), v2 v2 (x,t) are complex-valued potential functions, ξ is a complex spectral parameter, i= −1 The relation between this 3rd-order complex spectral problem and the associated completely integrable system is considered. Each equation governs the evolution of one of the components of the field transverse to the direction of propagation 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 [5]. This system is widely studied [6] [7] and used as a key model in the field of optical solitons in fibers [8] [9] 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 [10]-[12]
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