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Applications of the weighted scheme for GNLS equations in solving soliton solutions. (English) Zbl 0918.35130

The authors study soliton solutions of the generalized nonlinear Schrödinger equation (GNLS), which describes the multiple-photon absorption in optical fibers with slowly decreasing dispersion. The cases when the nonlinear parameter \(\alpha\to \infty\) and \(\alpha\to 0\) are also discussed. The soliton solutions are investigated analytically and numerically. A transvelling wave method is used to define one-soliton solutions for the generalized nonlinear Schrödinger equation. Then a finite difference method is applied to solve this equation numerically. In 2-D space a difference method is given by using a six-point weighted scheme, and its convergence and stability are studied. The numerical and analytical results are compared. Finally, conditions for the existence of bright soliton and dark soliton of the hyperbolic type are given.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
78A60 Lasers, masers, optical bistability, nonlinear optics
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