Difference between revisions of "Cubic NLS on T"

Cubic NLS on ${\displaystyle \mathbb {T} }$
Description
Equation ${\displaystyle iu_{t}+u_{xx}=\pm |u|^{2}u}$
Fields ${\displaystyle u:\mathbb {R} \times \mathbb {T} \to \mathbb {C} }$
Data class ${\displaystyle u(0)\in H^{s}(\mathbb {T} )}$
Basic characteristics
Structure completely integrable
Nonlinearity semilinear
Linear component Schrodinger
Critical regularity ${\displaystyle {\dot {H}}^{-1/2}(\mathbb {R} )}$
Criticality mass-subcritical;
energy-subcritical
Covariance Galilean
Theoretical results
LWP ${\displaystyle H^{s}(\mathbb {T} )}$ for ${\displaystyle s\geq 0}$
GWP ${\displaystyle H^{s}(\mathbb {T} )}$ for ${\displaystyle s\geq 0}$
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 ${\displaystyle s\geq 0\,}$ Bo1993.
• For ${\displaystyle s<0\,}$ one has failure of uniform local well-posedness CtCoTa-p, BuGdTz-p. In fact, the solution map is not even continuous from ${\displaystyle H^{s}\,}$ to ${\displaystyle H^{\sigma }\,}$ for any ${\displaystyle \sigma }$, even for small times and small data CtCoTa-p3.
• GWP for ${\displaystyle s\geq 0\,}$ thanks to ${\displaystyle L^{2}\,}$ conservation Bo1993.
• One also has GWP for random data whose Fourier coefficients decay like ${\displaystyle 1/|k|\,}$ (times a Gaussian random variable) Bo1995c. Indeed one has an invariant measure.
• If the cubic non-linearity is of ${\displaystyle {\underline {uuu}}\,}$ type (instead of ${\displaystyle |u|^{2}u\,}$) then one can obtain LWP for ${\displaystyle s>-1/3\,}$ 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.