# Difference between revisions of "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.