Split-step orthogonal spline collocation method for the complex Ginzburg-Landau equation in two dimensions

2017 ◽  
Author(s):  
Shanshan Wang
Keyword(s):  
Collocation Method ◽  
Landau Equation ◽  
Two Dimensions ◽  
Spline Collocation ◽  
Ginzburg Landau ◽  
Spline Collocation Method ◽  
Orthogonal Spline Collocation ◽  
Ginzburg Landau Equation

Dark soliton solutions of the cubic-quintic complex Ginzburg–Landau equation with high-order terms and potential barriers in normal-dispersion fiber lasers

2021 ◽  
Author(s):  
Marco A. Viscarra ◽  
Deterlino Urzagasti
Keyword(s):  
Optical Fibers ◽  
Landau Equation ◽  
Soliton Solutions ◽  
Dark Soliton ◽  
Two Dimensions ◽  
Ginzburg Landau ◽  
Normal Dispersion ◽  
Homogeneous Solutions ◽  
Dark Solitons ◽  
Ginzburg Landau Equation

In this paper, we numerically study dark solitons in normal-dispersion optical fibers described by the cubic-quintic complex Ginzburg–Landau equation. The effects of the third-order dispersion, self-steepening, stimulated Raman dispersion, and external potentials are also considered. The existence, chaotic content and interactions of these objects are analyzed, as well as the tunneling through a potential barrier and the formation of dark breathers aside from dark solitons in two dimensions and their mutual interactions as well as with periodic potentials. Furthermore, the homogeneous solutions of the model and the conditions for their stability are also analytically obtained.

scholarly journals Crank-Nicolson orthogonal spline collocation method combined with WSGI difference scheme for the two-dimensional time-fractional diffusion-wave equation

2020 ◽  
Vol 18 (1) ◽  
pp. 67-86
Author(s):  
Xiaoyong Xu ◽  
Fengying Zhou
Keyword(s):  
Collocation Method ◽  
Wave Equations ◽  
Stability And Convergence ◽  
Spline Collocation ◽  
Spline Collocation Method ◽  
Orthogonal Spline Collocation ◽  
Fractional Wave Equations ◽  
Discrete Orthogonal ◽  
The Stability ◽  
Crank Nicolson

Abstract In this paper, a discrete orthogonal spline collocation method combining with a second-order Crank-Nicolson weighted and shifted Grünwald integral (WSGI) operator is proposed for solving time-fractional wave equations based on its equivalent partial integro-differential equations. The stability and convergence of the schemes have been strictly proved. Several numerical examples in one variable and in two space variables are given to demonstrate the theoretical analysis.

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