Cubic NLS on T

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Cubic NLS on
Description
Equation
Fields
Data class
Basic characteristics
Structure completely integrable
Nonlinearity semilinear
Linear component Schrodinger
Critical regularity
Criticality mass-subcritical;
energy-subcritical
Covariance Galilean
Theoretical results
LWP for
GWP for
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 one has failure of uniform local well-posedness CtCoTa-p, BuGdTz-p. In fact, the solution map is not even continuous from to for any , 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 (times a Gaussian random variable) Bo1995c. Indeed one has an invariant measure.
  • If the cubic non-linearity is of type (instead of ) then one can obtain LWP for 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.