Cubic NLS

From DispersiveWiki

**Cubic NLS**
Description
Equation	$iu_t + \Delta u = \pm \|u\|^2 u$
Fields	$u: \R \times \R^d \to \mathbb{C}$
Data class	$u(0) \in H^s(\R^d)$
Basic characteristics
Structure	Hamiltonian
Nonlinearity	semilinear
Linear component	Schrodinger
Critical regularity	$\dot H^{d/2 - 1}(\R^d)$
Criticality	varies
Covariance	Galilean
Theoretical results
LWP	$H^s(\R^d)$ for $s \geq \max(d/2-1, 0)$
GWP	varies
Related equations
Parent class	NLS
Special cases	on R, on T, on R^2, on T^2, on R^3, on R^4
Other related	Schrodinger maps, mKdV, Zakharov

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 $R d$ , 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 $\pm |u|^2 u$ is replaced by a non-Hamiltonian, non-Galilean-invariant cubic polynomial such as $u 3$ or $\overline{u}^3$ . 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.

[edit]

Scaling analysis

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

$u(t,x) \mapsto \frac{1}{\lambda} u(\frac{t}{\lambda^2}, \frac{x}{\lambda}).$

Thus the critical regularity is $s_c = \frac{d}{2} - 1$ .

[edit]

Specific domains

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)