The Nonlinear Schrödinger Equation
SelfFocusing and Wave Collapse
Creator  
Publisher 
19990618

ISBN  0387227687, 9780387227689, 0387986111, 9780387986111,

Language 
English

Category  
Subject  Nonlinear theories. Nonlinear theories.  fast  (OCoLC)fst01038812 Schrödinger equation. Schrödinger equation.  fast  (OCoLC)fst01108121 Schrödinger, Equation de.  ram Schrödinger, Équation de. SCIENCE  Waves & Wave Mechanics.  bisacsh Théories non linéaires. Théories non linéaires.  ram 
Description
Regardez les singularit ́ es : il n’y a que ca ̧ qui compte. Gaston Julia The nonlinear Schr ̈ odinger (NLS) equation provides a canonical descr tion for the envelope dynamics of a quasimonochromatic plane wave (the carrying wave) propagating in a weakly nonlinear dispersive medium when dissipative processes are negligible. On short times and small propagation distances, the dynamics are linear, but cumulative nonlinear interactions result in a signi?cant modulation of the wave amplitude on large spatial and temporal scales. The NLS equation expresses how the linear dispersion relation is a?ected by the thickening of the spectral lines associated to the modulationandtheresonantnonlinearinteractions.Inoptics,itcanalsobe viewed as the extension to nonlinear media of the paraxial approximation, extensively used for linear waves propagating in random media. The NLS equation assumes weak nonlinearities but a ?nite dispersion at the scale of the carrying wave, while in situations where both dispersion and nonlinearities are equally weak, a “reductive perturbative exp sion” leads to longwavelength equations, like the Korteweg–de Vries, the Benjamin–Onoor,inseveraldimensions,theKadomtsev–Petviashvilieq tions (Segur 1978, Ablowitz and Segur 1981). This class of equations also includes the socalled derivative nonlinear Schr ̈ odinger (DNLS) equation obeyed by dispersive Alfv ́ en waves propagating along an ambient magnetic ?eld in a quasineutral plasma, because of a phasevelocity degeneracy in the dispersionless limit (see Mjølhus and Hada 1997 for a recent review).