# Difference between revisions of "Cubic NLS on T"

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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]] | [[Category:Equations]] |

## Revision as of 04:55, 8 August 2006

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.