Cubic NLS

The cubic NLS equation

The cubic NLS is the equation

$iu_{t}+\Delta u=\pm |u|^{2}u$

where $u(t,x)$ is complex-valued. 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.

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 $s_{c}={\frac {d}{2}}-1$.

Specific domains

• Cubic NLS on R (Mass and energy sub-critical; scattering-critical; completely integrable)
• 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 R x T (Mass-critical; energy-subcritical)
• Cubic NLS on T^2 (Mass-critical; energy-subcritical)
• Cubic NLS on R^3 (Mass-supercritical; energy-subcritical; scattering-subcritical)
• Cubic NLS on T^3 (Mass-supercritical; energy-subcritical)
• Cubic NLS on R^4 (Mass-supercritical; energy-critical; scattering-subcritical)
• Cubic NLS on T^4 (Mass-supercritical; energy-critical)
• Cubic NLS on S^6 (Mass-supercritical; energy-supercritical)