Abstract
This paper discusses approaches to the numerical integration of the coupled nonlinear Schrödinger equations system, different from the generally accepted approach based on the method of splitting according to physical processes. A combined explicit/implicit finite-difference integration scheme based on the implicit Crank–Nicolson finite-difference scheme is proposed and substantiated. It allows the integration of a nonlinear system of equations with a choice of nonlinear terms from the previous integration step. The main advantages of the proposed method are: its absolute stability through the use of an implicit finite-difference integration scheme and an integrated mechanism for refining the numerical solution at each step; integration with automatic step selection; performance gains (or resolutions) up to three or more orders of magnitude due to the fact that there is no need to produce direct and inverse Fourier transforms at each integration step, as is required in the method of splitting according to physical processes. An additional advantage of the proposed method is the ability to calculate the interaction with an arbitrary number of propagation modes in the fiber.
Highlights
Femtosecond lasers hold a strong position in the current industrial production of materials and different-purpose products [1,2,3,4]
Coupled nonlinear Schrödinger Equations System for two orthogonally polarized modes (AX and AY ), propagating in a birefringent fiber, which is used for modeling of short optical pulses propagation, has a form equivalent to Equation (1):
The number of mesh points along dimensionless time was chosen as 20,000; approximately 720,000 integration steps were made along dimensionless time with initial time step ∆ξ = 1·10−4 d.u
Summary
Femtosecond lasers hold a strong position in the current industrial production of materials and different-purpose products [1,2,3,4]. Linear terms are written in an implicit scheme, and nonlinear terms in an explicit finite-difference scheme This approach allows researchers to divide the system of Schrödinger equations into two independent systems of linear equations for each mode at each step of numerical integration process. The main advantages of the proposed method are the following: absolute stability due to the usage of an implicit finite-difference integration scheme and an integrated mechanism for refining the numerical solution at each step; integration with automatic step selection; increase in the efficiency (or resolutions) up to three or more orders of magnitude due to the fact that there is no need to produce direct and inverse Fourier transforms at each integration step, as is required in a split-step method. An additional advantage of the proposed method is the ability to calculate the interaction with an arbitrary number of propagation modes in the fiber
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