Geometric Optics and Long Range Scattering¶for One-Dimensional Nonlinear Schrödinger Equations

2001 ◽  
Vol 220 (1) ◽  
pp. 41-67 ◽  
Author(s):  
Rémi Carles
Keyword(s):  
Long Range ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Geometric Optics ◽  
One Dimensional ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations

scholarly journals NUMERICAL ASPECTS OF NONLINEAR SCHRÖDINGER EQUATIONS IN THE PRESENCE OF CAUSTICS

2007 ◽  
Vol 17 (10) ◽  
pp. 1531-1553 ◽  
Author(s):  
RÉMI CARLES ◽  
LAURENT GOSSE
Keyword(s):  
Operator Theory ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Scattering Operator ◽  
Rigorous Results ◽  
One Dimensional ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Focus Point

The aim of this paper is to develop on the asymptotics of some one-dimensional nonlinear Schrödinger equations from both the theoretical and the numerical perspectives, when a caustic is formed. We review rigorous results in the field and give some heuristics in cases where justification is still needed. The scattering operator theory is recalled. Numerical experiments are carried out on the focus point singularity for which several results have been proved rigorously. Furthermore, the scattering operator is numerically studied. Finally, experiments on the cusp caustic are displayed, and similarities with the focus point are discussed. Several shortcomings of spectral time-splitting schemes are investigated.

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