Cubic NLS on T: Difference between revisions

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{{equation
| name = Cubic NLS on <math>\mathbb{T}</math>
| equation = <math>iu_t + u_{xx} = \pm |u|^2 u</math>
| fields = <math>u: \R \times \mathbb{T} \to \mathbb{C}</math>
| data = <math>u(0) \in H^s(\mathbb{T})</math>
| hamiltonian = [[completely integrable]]
| linear = [[free Schrodinger equation|Schrodinger]]
| nonlinear = [[semilinear]]
| critical = <math>\dot H^{-1/2}(\R)</math>
| criticality = mass-subcritical;<br> energy-subcritical
| covariance = [[Galilean]]
| lwp = <math>H^s(\mathbb{T})</math> for <math>s \geq 0</math>
| gwp = <math>H^s(\mathbb{T})</math> for <math>s \geq 0</math>
| parent = [[cubic NLS]]
| special = -
| related = [[KdV on T|KdV]], [[mKdV on T|mKdV]]
}}
The theory of the [[cubic NLS]] on the circle is as follows.
The theory of the [[cubic NLS]] on the circle is as follows.


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** One also has GWP for random data whose Fourier coefficients decay like <math>1/|k|\,</math> (times a Gaussian random variable) [[Bo1995c]]. Indeed one has an invariant measure.
** One also has GWP for random data whose Fourier coefficients decay like <math>1/|k|\,</math> (times a Gaussian random variable) [[Bo1995c]]. Indeed one has an invariant measure.
* If the cubic non-linearity is of <math>\underline{uuu}\,</math> type (instead of <math>|u|^2u\,</math>) then one can obtain LWP for <math>s > -1/3\,</math> [[Gr-p2]]
* If the cubic non-linearity is of <math>\underline{uuu}\,</math> type (instead of <math>|u|^2u\,</math>) then one can obtain LWP for <math>s > -1/3\,</math> [[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.
* ''Remark'': This equation is [[completely integrable]] [[AbMa1981]]; all higher order integer Sobolev norms stay bounded. Growth of fractional norms might be interesting, though.


[[Category:Integrability]]
[[Category:Integrability]]
[[Category:Schrodinger]]
[[Category:Schrodinger]]
[[Category:Equations]]

Latest revision as of 05:06, 8 August 2006

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.