Coupled nonlinear Schrödinger equations in optic fibers theory

In this paper a detailed derivation and numerical solutions of Coupled Nonlinear Schrödinger Equations for pulses of polarized electromagnetic waves in cylindrical fibers has been reviewed. Our recent work has been compared with some previous ones and the advantage of our new approach over other methods has been assessed. The novelty of our approach lies is an attempt to proceed without loss of information within the frame of basic approximations. In our work we focused on the multimode The eigen mode definition is based on complete linearized Maxwell equations and Hondros-Debye boundary conditions, which depend on the geometry of the dielectric waveguide. We proved both stability and convergence in the L2 space of an explicit finite-difference scheme for the Coupled Nonlinear Schrödinger Equations and those estimations are used for an implicit scheme. To test our hypothesis we compare numerical solutions for Manakov system with known analytical solitonic solutions. We also consider an important example of the general system - an evolution of two pulses with different group velocity which can serve as a model of pulses interaction in multimode optic fibers. Last case, a nonlinear dispersion of rectangular pulse, exhibits an asymptotic behavior similar to Nonlinear Schrödinger Equation solution asymptotics for the rectangular initial condition. Finally, we compared theoretical results with specially arranged experiments employing a photonic crystal fiber