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| ==Overview== | | == Brain Teasers - How To Start A Good Story == |
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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.
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| One can generalize both the linear and nonlinear perturbations to these equations and consider
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| 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.
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| 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]].
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| 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]].
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| ==Specific Schrodinger Equations==
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| Monomial [[semilinear Schrodinger equation]]s can indexed by the degree of the nonlinearity, as follows.
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| ===Quadratic NLS===
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| [[NLS]] equations of the form
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| <math> i \partial_t u + \Delta u = Q(u, \overline{u})</math>
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| 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.
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| ===Cubic NLS===
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| The [[cubic NLS|cubic nonlinear Schrodinger equation]] is of the form
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| <math> i \partial_t u + \Delta u = \pm |u|^2 u</math>
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| They are [[completely integrable]] in one dimension, mass-critical in two-dimensions, and energy-critical in four dimensions.
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| ===Quartic NLS===
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| A [[NLS|nonlinear Schrodinger equation]] with nonlinearity of degree 4 is a [[quartic NLS|quartic nonlinear Schrodinger equation]].
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| ===Quintic NLS===
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| [[NLS]] equations of the form
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| <math> i \partial_t u + \Delta u = \pm |u|^4 u</math>
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| are [[quintic NLS|quintic nonlinear Schrodinger equations]]. They are mass-critical in one dimension and energy-critical in three dimensions.
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| ===Septic NLS===
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| [[NLS]] equations of the form
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| <math> i \partial_t u + \Delta u = \pm |u|^6 u</math>
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| are [[septic NLS|septic nonlinear Schrodinger equations]].
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| ===<math>L^2</math>-critical NLS===
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| The [[NLS|nonlinear Schrodinger equation]]
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| <math> i \partial_t u + \Delta u = \pm |u|^{\frac{4}{d}} u</math>
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| 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]].
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| ===Higher order NLS===
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| 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]].
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| ===Schrodinger maps===
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| 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]].
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| ===Cubic DNLS on <math>R</math>===
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| The [[cubic DNLS on R|deriviative cubic nonlinear Schrodinger equation]] has nonlinearity of the form <math>i \partial_x (|u|^2 u).</math>
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| ===Hartree Equation===
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| 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]].
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| ===Maxwell-Schrodinger system===
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| A Schrodinger-wave system closely related to the [[Maxwell-Klein-Gordon equation]] is the [[Maxwell-Schrodinger system]].
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| [[Category:Schrodinger]]
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| [[Category:Equations]]
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