Difference between revisions of "Cubic NLS on T"

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  | nonlinear = [[semilinear]]
  | nonlinear = [[semilinear]]
  | critical = <math>\dot H^{-1/2}(\R)</math>
  | critical = <math>\dot H^{-1/2}(\R)</math>
  | criticality = mass-subcritical;<br> energy-subcritical;<br> scattering-critical
  | criticality = mass-subcritical;<br> energy-subcritical
  | covariance = [[Galilean]]
  | covariance = [[Galilean]]
  | lwp = <math>H^s(\mathbb{T})</math> for <math>s \geq 0</math>
  | lwp = <math>H^s(\mathbb{T})</math> for <math>s \geq 0</math>

Latest revision as of 05:06, 8 August 2006

Cubic NLS on
Data class
Basic characteristics
Structure completely integrable
Nonlinearity semilinear
Linear component Schrodinger
Critical regularity
Criticality mass-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.