# Cubic NLS

Cubic NLS
Description
Equation ${\displaystyle iu_{t}+\Delta u=\pm |u|^{2}u}$
Fields ${\displaystyle u:\mathbb {R} \times \mathbb {R} ^{d}\to \mathbb {C} }$
Data class ${\displaystyle u(0)\in H^{s}(\mathbb {R} ^{d})}$
Basic characteristics
Structure Hamiltonian
Nonlinearity semilinear
Linear component Schrodinger
Critical regularity ${\displaystyle {\dot {H}}^{d/2-1}(\mathbb {R} ^{d})}$
Criticality varies
Covariance Galilean
Theoretical results
LWP ${\displaystyle H^{s}(\mathbb {R} ^{d})}$ for ${\displaystyle s\geq \max(d/2-1,0)}$
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 ${\displaystyle R^{d}}$, 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 ${\displaystyle \pm |u|^{2}u}$ is replaced by a non-Hamiltonian, non-Galilean-invariant cubic polynomial such as ${\displaystyle u^{3}}$ or ${\displaystyle {\overline {u}}^{3}}$. 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.

## Scaling analysis

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

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

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