The cubic NLS is displayed on the box on the right. 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.
One can also consider variants of the cubic NLS in which the (Hamiltonian, Galilean-invariant) nonlinearity is replaced by a non-Hamiltonian, non-Galilean-invariant cubic polynomial such as or . Typically, for this variant the local theory remains unchanged (or even improves somewhat), but the global theory is lost (especially for large data) due to the lack of conservation laws.
On Euclidean domains at least, the cubic NLS obeys the scale invariance
Thus the critical regularity is .
- Cubic NLS on R (Mass and energy sub-critical; scattering-critical; completely integrable)
- Cubic NLS on the half-line and interval (Mass and energy sub-critical)
- Cubic NLS on T (Mass and energy sub-critical; completely integrable)
- Cubic NLS on R^2 (Mass-critical; energy-subcritical; scattering-subcritical)
- Cubic NLS on two-dimensional manifolds (Mass-critical; energy-subcritical)
- Cubic NLS on R^3 (Mass-supercritical; energy-subcritical; scattering-subcritical)
- Cubic NLS on three-dimensional manifolds (Mass-supercritical; energy-subcritical)
- Cubic NLS on R^4 (Mass-supercritical; energy-critical; scattering-subcritical)
- Cubic NLS on four-dimensional manifolds (Mass-supercritical; energy-critical)
- Cubic NLS on six-dimensional manifolds (Mass-supercritical; energy-supercritical)