Integrability of an $N$-coupled nonlinear Schr\"{o}dinger system for polarized optical waves in an isotropic medium via symbolic computation
Hai-Qiang Zhang, Tao Xu, Juan Li, and Bo Tian

Considering the simultaneous propagation of multi-component fields in an isotropic medium, an {\em N}-coupled nonlinear Schr\"{o}dinger system with the self-phase modulation, cross-phase modulation and energy exchange terms is investigated in this paper. Firstly, via symbolic computation, the Painlev\'{e} singularity structure analysis shows that such a system admits the Painlev\'{e} property. Then, with the Ablowitz-Kaup-Newell-Segur scheme, the linear eigenvalue problem (Lax pair) associated with this model is constructed in the frame of the block matrices. With the Hirota bilinear method, the bright one- and two-soliton solutions of this system are presented. In addition, the bright multi-soliton solutions of the system for $N=2$ are straightforwardly derived by the linear superposition of soliton solutions of two independent scalar nonlinear Schr\"{o}dinger equations. Furthermore, through the analysis for the soliton solutions, the corresponding propagation behavior and applications for soliton pulses in nonlinear optical fibers are considered. Finally, three significant conserved quantities, i.e., energy, momentum and Hamiltonian, are also given. \\ \\ {\em PACS numbers:} 05.45.Yv; 02.30.Ik; 42.65.Tg; 02.70.Wz

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