View source for Quintic NLS on R

The local and global theory of the quintic NLS on <math>R</math> is as follows.

* This equation may be viewed as a simpler version of [[cubic DNLS on R|cubic DNLS]], and is always at least as well-behaved. It has been proposed as a modifiation of the Gross-Pitaevski approximation for low-dimesional Bose liquids [[KolNewStrQi2000]]
* Scaling is <math>s_c = 0\,</math>, thus this is an [[mass critical NLS]].
* LWP is known for <math>s \ge 0\,</math> [[CaWe1990]], [[Ts1987]].
** For <math>s=0\,</math> the time of existence depends on the profile of the data as well as the norm.
** Below <math>L^2\,</math> we have ill-posedness by Gallilean invariance considerations in both the focusing [[KnPoVe-p]] and defocusing [[CtCoTa-p2]] cases.
* GWP for <math>s>4/9\,</math> in the defocussing case [[Tzi-p]]
** For <math>s>1/2\,</math> this is in [[CoKeStTkTa2002b]]
** For <math>s>2/3\,</math> this is in [[CoKeStTkTa2001b]].
** For <math>s > 32/33\,</math> this is implicit in [[Tk-p]].
** For <math>s\ge 1\,</math> this follows from LWP and Hamiltonian conservation.
** One has GWP and scattering for small <math>L^2\,</math> data for any quintic non-linearity. The corresponding problem for large <math>L^2\,</math> data and defocussing nonlinearity is very interesting, but probably very difficult, perhaps only marginally easier than the corresponding problem for the [[cubic NLS|2D cubic NLS]]. It would suffice to show that the solution has a bounded <math>L^6\,</math> norm in spacetime.
** Explicit blowup solutions (with large <math>L^2\,</math> norm) are known in the focussing case [[BirKnPoSvVe1996]]. The blowup rate in <math>H^1\,</math> is <math>t^{-1}\,</math> in these solutions. This is not the optimal blowup rate; in fact an example has been constructed where the blowup rate is <math>|t|^{-1/2} (log log|t|)^{1/2}\,</math>[[Per-p]]. Furthermore, one always this blowup behavior (or possibly slower, though one must still blow up by at least <math>|t|^{-1/2}\,</math>) whenever the energy is negative [[MeRap-p]], [[MeRap-p2]], and one either assumes that the mass is close to the critical mass or that <math>xu\,</math> is in <math>L^2\,</math>.
*** One can modify the explicit solutions from [[BirKnPoSvVe1996]] and in fact create solutions which blow up at any collection of specified points in spacetime [[BoWg1997]], [[Nw1998]].
* ''Remark'': This equation is invariant under the [[pseudoconformal transformation]]. GWP results in <math>H^s\,</math> automatically transfer to GWP and scattering results in <math>L^2(|x|^s)\,</math> thanks to the pseudo-conformal transformation.
* Solitons are <math>H^1\,</math>-unstable.

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