Schrodinger equations: Difference between revisions

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==Overview==
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There are many '''nonlinear Schrodinger equations''' in the literature, all of which are perturbations of one sort or another of the [[free Schrodinger equation]]. One general class of such equations takes the form
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<center><math>i \partial_t u + \Delta u = f (u, \overline{u}, Du, D \overline{u})</math></center>
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where <math>D</math> denotes spatial differentiation. In such full generality, we refer to this equation as a [[derivative non-linear Schrodinger equation]] (D-NLS). If the non-linearity does not contain derivatives then we refer to this equation as a [[semilinear Schrodinger equation]] (NLS). These equations (particularly the [[cubic NLS]]) arise as model equations from several areas of physics.
 
One can generalize both the linear and nonlinear perturbations to these equations and consider
the class of [[quasilinear Schrodinger equations]] or even [[fully nonlinear Schrodinger equations]]. Needless to say, these equations are significantly more difficult to analyse than the simpler model cases discussed above.
 
One can combine these nonlinear perturbations with a [[free Schrodinger equation|linear perturbation]], leading for instance to the [[NLS with potential]] and the [[NLS on manifolds and obstacles]].
 
The perturbative theory of nonlinear Schrodinger equations (and the [[NLS|semilinear Schrodinger equations]] in particular) rests on a number of [[Schrodinger estimates|linear and nonlinear estimates for the free Schrodinger equation]].
 
 
==Specific Schrodinger Equations==
 
Monomial [[semilinear Schrodinger equation]]s can indexed by the degree of the nonlinearity, as follows.
 
===Quadratic NLS===
 
[[NLS]] equations of the form
 
<math> i \partial_t u + \Delta u = Q(u, \overline{u})</math>
 
with <math>Q(u, \overline{u})</math> a quadratic function of its arguments are [[quadratic NLS|quadratic nonlinear Schrodinger equations]].  They are mass-critical in four dimensions.
 
===Cubic NLS===
 
The [[cubic NLS|cubic nonlinear Schrodinger equation]] is of the form
 
<math> i \partial_t u + \Delta u = \pm |u|^2 u</math>
 
They are [[completely integrable]] in one dimension, mass-critical in two-dimensions, and energy-critical in four dimensions.
 
===Quartic NLS===
 
A [[NLS|nonlinear Schrodinger equation]] with nonlinearity of degree 4 is a [[quartic NLS|quartic nonlinear Schrodinger equation]].
 
===Quintic NLS===
 
[[NLS]] equations of the form
 
<math> i \partial_t u + \Delta u = \pm |u|^4 u</math>
 
are [[quintic NLS|quintic nonlinear Schrodinger equations]].  They are mass-critical in one dimension and energy-critical in three dimensions.
 
===Septic NLS===
 
[[NLS]] equations of the form
 
<math> i \partial_t u + \Delta u = \pm |u|^6 u</math>
 
are [[septic NLS|septic nonlinear Schrodinger equations]].
 
===<math>L^2</math>-critical NLS===
 
The [[NLS|nonlinear Schrodinger equation]]
 
<math> i \partial_t u + \Delta u = \pm |u|^{\frac{4}{d}} u</math>
 
posed for <math>x \in R^d</math> is scaling invariant in <math>L^2_x</math>. This family of nonlinear Schrodinger equations is therefore called the [[mass critical NLS|mass critical nonlinear Schrodinger equation]].
 
===Higher order NLS===
 
One can study higher-order NLS equations in which the Laplacian is replaced by a higher power. One class of such examples comes from the infinite hierarchy of commuting flows arising from the completely integrable [[cubic NLS]] on <math>R</math>.  Another is the [[nonlinear Schrodinger-Airy system]].  A third class arises from the elliptic case of the [[Zakharov-Schulman system]].
 
===Schrodinger maps===
 
A geometric [[derivative non-linear Schrodinger equation]] that has been intensively studied is the [[Schrodinger maps|Schrodinger map equation]].  This is the Schrodinger counterpart of the [[wave maps equation]].
 
===Cubic DNLS on <math>R</math>===
 
The [[cubic DNLS on R|deriviative cubic nonlinear Schrodinger equation]] has nonlinearity of the form <math>i \partial_x (|u|^2 u).</math> 
 
===Hartree Equation===
 
The [[Hartree equation]] has a nonlocal nonlinearity given by convolution, as does the very similar [[Schrodinger-Poisson system]], and certain cases of the [[Davey-Stewartson system]].
 
===Maxwell-Schrodinger system===
 
A Schrodinger-wave system closely related to the [[Maxwell-Klein-Gordon equation]] is the [[Maxwell-Schrodinger system]].
 
 
[[Category:Schrodinger]]
[[Category:Equations]]

Revision as of 19:14, 19 January 2011

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