Cubic NLS on T

**Cubic NLS on $\mathbb{T}$**
Description
Equation	$iu_t + u_{xx} = \pm \|u\|^2 u$
Fields	$u: \R \times \mathbb{T} \to \mathbb{C}$
Data class	$u(0) \in H^s(\mathbb{T})$
Basic characteristics
Structure	completely integrable
Nonlinearity	semilinear
Linear component	Schrodinger
Critical regularity	$\dot H^{-1/2}(\R)$
Criticality	mass-subcritical; energy-subcritical
Covariance	Galilean
Theoretical results
LWP	$H^s(\mathbb{T})$ for $s \geq 0$
GWP	$H^s(\mathbb{T})$ for $s \geq 0$
Related equations
Parent class	cubic NLS
Special cases	-
Other related	KdV, mKdV

The theory of the cubic NLS on the circle is as follows.

LWP for Bo1993.
- For $s<0\,$ one has failure of uniform local well-posedness CtCoTa-p, BuGdTz-p. In fact, the solution map is not even continuous from $H^s\,$ to $H^{\sigma}\,$ for any $σ$ , even for small times and small data CtCoTa-p3.
GWP for thanks to conservation Bo1993.
- One also has GWP for random data whose Fourier coefficients decay like $1/|k|\,$ (times a Gaussian random variable) Bo1995c. Indeed one has an invariant measure.
If the cubic non-linearity is of $\underline{uuu}\,$ type (instead of $|u|^2u\,$ ) then one can obtain LWP for $s > -1/3\,$ Gr-p2
Remark: This equation is completely integrable AbMa1981; all higher order integer Sobolev norms stay bounded. Growth of fractional norms might be interesting, though.