Difference between revisions of "Cubic NLS on T"

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