scholarly journals WKB-Based Schemes for the Oscillatory 1D Schrödinger Equation in the Semiclassical Limit

2011 ◽  
Vol 49 (4) ◽  
pp. 1436-1460 ◽  
Author(s):  
Anton Arnold ◽  
Naoufel Ben Abdallah ◽  
Claudia Negulescu
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Semiclassical Limit

scholarly journals Dispersion Lemmas Revisited

VLSI Design ◽  
1999 ◽  
Vol 9 (4) ◽  
pp. 365-375
Author(s):  
I. Gasser ◽  
P. A. Markowich ◽  
B. Perthame
Keyword(s):  
Dirac Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Semiclassical Limit ◽  
Kinetic Equations ◽  
Wigner Transform ◽  
Dirac Equations

We investigate regularizing dispersive effects for various classical equations, e.g., the Schrödinger and Dirac equations. After Wigner transform, these dispersive estimates are reduced to moment lemmas for kinetic equations. They yield new regularization results for the Schrödinger equation (valid up to the semiclassical limit) and the Dirac equation.

scholarly journals Large Time-Stepping Spectral Methods for the Semiclassical Limit of the Defocusing Nonlinear Schrödinger Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-27
Author(s):  
Zongqi Liang
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Spectral Methods ◽  
Schrodinger Equation ◽  
Large Time ◽  
Semiclassical Limit ◽  
Nonlinear Schrodinger Equation ◽  
Time Step ◽  
Nonlinear Schrödinger ◽  
Time Stepping

We analyze a class of large time-stepping Fourier spectral methods for the semiclassical limit of the defocusing Nonlinear Schrödinger equation and provide highly stable methods which allow much larger time step than for a standard implicit-explicit approach. An extra term, which is consistent with the order of the time discretization, is added to stabilize the numerical schemes. Meanwhile, the first-order and second-order semi-implicit schemes are constructed and analyzed. Finally the numerical experiments are performed to demonstrate the effectiveness of the large time-stepping approaches.

scholarly journals Uniformly accurate time-splitting methods for the semiclassical linear Schrödinger equation

2019 ◽  
Vol 53 (2) ◽  
pp. 443-473 ◽  
Author(s):  
Philippe Chartier ◽  
Loïc Le Treust ◽  
Florian Méhats
Keyword(s):  
Numerical Methods ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Semiclassical Limit ◽  
Splitting Methods ◽  
Convergence Behavior ◽  
First Order ◽  
Time Splitting ◽  
Uniform Accuracy

This article is devoted to the construction of numerical methods which remain insensitive to the smallness of the semiclassical parameter for the linear Schrödinger equation in the semiclassical limit. We specifically analyse the convergence behavior of the first-order splitting. Our main result is a proof of uniform accuracy. We illustrate the properties of our methods with simulations.

SEMICLASSICAL LIMIT OF THE DERIVATIVE NONLINEAR SCHRÖDINGER EQUATION

2000 ◽  
Vol 10 (02) ◽  
pp. 261-285 ◽  
Author(s):  
BENOÎT DESJARDINS ◽  
CHI-KUN LIN ◽  
TAI-CHENG TSO
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Semiclassical Limit ◽  
Nonlinear Schrodinger Equation ◽  
Limit System ◽  
Nonlinear Schrödinger ◽  
Derivative Nonlinear Schrödinger Equation ◽  
Genuine Nonlinearity ◽  
Dispersion Limit

We study the semiclassical limit of the general derivative nonlinear Schrödinger equation for initial data with Sobolev regularity, before shocks appear in the limit system. The strict hyperbolicity and genuine nonlinearity is proved for the dispersion limit of the derivative nonlinear Schrödinger equation.

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