Multisymplecticity of Crank–Nicolson Scheme for the Nonlinear Schrödinger Equation

2002 ◽  
Vol 71 (9) ◽  
pp. 2348-2349 ◽  
Author(s):  
Jing-Bo Chen
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Nicolson Scheme ◽  
Crank Nicolson

scholarly journals Numerical Approximation of The Weakly Damped Nonlinear Schrödinger Equation

2001 ◽  
Vol 1 (4) ◽  
pp. 319-332 ◽  
Author(s):  
Raimondas Čiegis ◽  
Violeta Pakenienė
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Optical Fibers ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Stability And Convergence ◽  
Nonlinear Schrödinger ◽  
The Stability ◽  
Nicolson Scheme ◽  
Crank Nicolson

AbstractIn this paper we consider the one-dimensional nonlinear Schrödinger equation. The equation includes an absorption term, and the solution is periodically amplified in order to compensate the lose of the energy. The problem describes propa- gation of a signal in optical fibers. In our previous work we proved that the well-known Crank—Nicolson scheme is unconditionally unstable for this problem. We present in this paper two finite difference approximations. The first one is given by a modified Crank—Nicolson scheme and the second one is obtained by a splitting scheme. The stability and convergence of these schemes are proved. The results of numerical exper- iments are presented and discussed.

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