Difference between revisions of "Cubic NLS on T"
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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]] | ||
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Latest revision as of 05:06, 8 August 2006
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.
- 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.