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.
    • 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.