Cubic NLS: Difference between revisions

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* [[Cubic NLS on 2d manifolds|Cubic NLS on two-dimensional manifolds]] (Mass-critical; energy-subcritical)
* [[Cubic NLS on 2d manifolds|Cubic NLS on two-dimensional manifolds]] (Mass-critical; energy-subcritical)
* [[Cubic NLS on R3|Cubic NLS on R^3]] (Mass-supercritical; energy-subcritical; scattering-subcritical)
* [[Cubic NLS on R3|Cubic NLS on R^3]] (Mass-supercritical; energy-subcritical; scattering-subcritical)
* [[Cubic NLS on T3|Cubic NLS on T^3]] (Mass-supercritical; energy-subcritical)
* [[Cubic NLS on T3|Cubic NLS on three-dimensional manifolds]] (Mass-supercritical; energy-subcritical)
* [[Cubic NLS on R4|Cubic NLS on R^4]] (Mass-supercritical; energy-critical; scattering-subcritical)
* [[Cubic NLS on R4|Cubic NLS on R^4]] (Mass-supercritical; energy-critical; scattering-subcritical)
* [[Cubic NLS on T4|Cubic NLS on T^4]] (Mass-supercritical; energy-critical)
* [[Cubic NLS on T4|Cubic NLS on four-dimensional manifolds]] (Mass-supercritical; energy-critical)
* [[Cubic NLS on S6|Cubic NLS on S^6]] (Mass-supercritical; energy-supercritical)
* [[Cubic NLS on S6|Cubic NLS on six-dimensional manifolds]] (Mass-supercritical; energy-supercritical)


[[Category:Equations]]
[[Category:Equations]]
[[Category:Schrodinger]]
[[Category:Schrodinger]]

Revision as of 21:22, 5 August 2006

The cubic NLS equation

The cubic NLS is the NLS equation

where is complex-valued. The sign + is defocusing, while the - sign is focusing. This equation is traditionally studied on Euclidean domains , but other domains are certainly possible.

In one spatial dimension the cubic NLS equation is completely integrable. but this is not the case in higher dimensions.

The cubic NLS can be viewed as an oversimplified model of the Schrodinger map equation. It also arises as the limit of a number of other equations, such as the modified Korteweg-de Vries equation and Zakharov system.

Scaling analysis

On Euclidean domains at least, the cubic NLS obeys the scale invariance

Thus the critical regularity is .

Specific domains