scholarly journals Semiclassical Limit of the Scattering Transform for the Focusing Nonlinear Schrodinger Equation

2011 ◽  
Author(s):  
A. Tovbis ◽  
S. Venakides
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Semiclassical Limit ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Scattering Transform

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.

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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