scholarly journals On Fourier time-splitting methods for nonlinear Schrödinger equations in the semi-classical limit II. Analytic regularity

2016 ◽  
Vol 136 (1) ◽  
pp. 315-342 ◽  
Author(s):  
Rémi Carles ◽  
Clément Gallo
Keyword(s):  
Classical Limit ◽  
Splitting Methods ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Time Splitting ◽  
Analytic Regularity

scholarly journals WKB analysis of generalized derivative nonlinear Schrödinger equations without hyperbolicity

2017 ◽  
Vol 27 (09) ◽  
pp. 1727-1742 ◽  
Author(s):  
Rémi Carles ◽  
Clément Gallo
Keyword(s):  
Classical Limit ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Polynomial Nonlinearity ◽  
Nonlinear Schrodinger Equations ◽  
Phase Amplitude ◽  
Analytic Regularity ◽  
Amplitude System

We consider the semi-classical limit of nonlinear Schrödinger equations in the presence of both a polynomial nonlinearity and the derivative in space of a polynomial nonlinearity. By working in a class of analytic initial data, we do not have to assume any hyperbolic structure on the (limiting) phase/amplitude system. The solution, its approximation, and the error estimates are considered in time-dependent analytic regularity.

scholarly journals WKB analysis of nonelliptic nonlinear Schrödinger equations

2019 ◽  
Vol 22 (06) ◽  
pp. 1950045 ◽  
Author(s):  
Rémi Carles ◽  
Clément Gallo
Keyword(s):  
Nonlinear Schrödinger Equations ◽  
Order System ◽  
Energy Estimates ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Leading Order ◽  
Qualitative Information ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Analytic Regularity

We justify the WKB analysis for generalized nonlinear Schrödinger equations (NLS), including the hyperbolic NLS and the Davey–Stewartson II system. Since the leading order system in this analysis is not hyperbolic, we work with analytic regularity, with a radius of analyticity decaying with time, in order to obtain better energy estimates. This provides qualitative information regarding equations for which global well-posedness in Sobolev spaces is widely open.

scholarly journals Efficient time-splitting Hermite-Galerkin spectral method for the coupled nonlinear Schrödinger equations

2020 ◽  
Vol 14 ◽  
pp. 174830262097353
Author(s):  
Qingqu Zhuang ◽  
Yi Yang
Keyword(s):  
Splitting Method ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Coupled Nonlinear Schrödinger Equations ◽  
Time Step ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Time Splitting ◽  
Coupled Nonlinear Schrodinger Equations

The paper focuses on efficient time-splitting Hermite-Galerkin spectral approximation of the coupled nonlinear Schrödinger equations on the whole line. The original problem is decomposed into one nonlinear subproblem and one linear subproblem by time-splitting method. At each time step, the nonlinear subproblem is solved exactly. While the linear subproblem is efficiently solved by choosing suitable Hermite basis functions with matrix decomposition technique. Numerical experiments are carried out to demonstrate the effectiveness and efficiency of the proposed method.

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