Cubic NLS on T

Cubic NLS on $\mathbb {T}$

Description
Equation	$iu_{t}+u_{xx}=\pm |u|^{2}u$
Fields	$u:\mathbb {R} \times \mathbb {T} \to \mathbb {C}$
Data class	$u(0)\in H^{s}(\mathbb {T} )$
Basic characteristics
Structure	completely integrable
Nonlinearity	semilinear
Linear component	Schrodinger
Critical regularity	${\dot {H}}^{-1/2}(\mathbb {R} )$
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 $s\geq 0\,$ 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 $\sigma$, even for small times and small data CtCoTa-p3.
• GWP for $s\geq 0\,$ thanks to $L^{2}\,$ 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 $|u|^{2}u\,$) 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.