Efficient semi-implicit compact finite difference scheme for nonlinear Schrödinger equations on unbounded domain

2020 ◽  
Vol 153 ◽  
pp. 319-343
Author(s):  
Yunxia Hu ◽  
Hongwei Li ◽  
Ziwen Jiang
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Unbounded Domain ◽  
Finite Difference Scheme ◽  
Nonlinear Schrödinger Equations ◽  
Schrödinger Equations ◽  
Compact Finite Difference Scheme ◽  
Compact Finite Difference ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations

scholarly journals Original Solution of Coupled Nonlinear Schrödinger Equations for Simulation of Ultrashort Optical Pulse Propagation in a Birefringent Fiber

Fibers ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 34 ◽  
Author(s):  
Airat Zhavdatovich Sakhabutdinov ◽  
Vladimir Ivanovich Anfinogentov ◽  
Oleg Gennadievich Morozov ◽  
Vladimir Alexandrovich Burdin ◽  
Anton Vladimirovich Bourdine ◽  
...  
Keyword(s):  
Finite Difference ◽  
Nonlinear Schrödinger Equations ◽  
Integration Scheme ◽  
Integration Step ◽  
Schrödinger Equations ◽  
Coupled Nonlinear Schrödinger Equations ◽  
Physical Processes ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Implicit Finite Difference

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.

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