Quintic NLS on R: Difference between revisions
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** For <math>s\ge 1\,</math> this follows from LWP and Hamiltonian conservation. | ** 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. | ** 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>. | ** 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]]. | *** 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. | * ''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. | ||
Revision as of 00:18, 3 February 2007
The local and global theory of the quintic NLS on is as follows.
- This equation may be viewed as a simpler version of 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 , thus this is an mass critical NLS.
- LWP is known for CaWe1990, Ts1987.
- GWP for in the defocussing case Tzi-p
- For this is in CoKeStTkTa2002b
- For this is in CoKeStTkTa2001b.
- For this is implicit in Tk-p.
- For this follows from LWP and Hamiltonian conservation.
- One has GWP and scattering for small data for any quintic non-linearity. The corresponding problem for large data and defocussing nonlinearity is very interesting, but probably very difficult, perhaps only marginally easier than the corresponding problem for the 2D cubic NLS. It would suffice to show that the solution has a bounded norm in spacetime.
- Explicit blowup solutions (with large norm) are known in the focussing case BirKnPoSvVe1996. The blowup rate in is in these solutions. This is not the optimal blowup rate; in fact an example has been constructed where the blowup rate is Per-p. Furthermore, one always this blowup behavior (or possibly slower, though one must still blow up by at least ) whenever the energy is negative MeRap-p, MeRap-p2, and one either assumes that the mass is close to the critical mass or that is in .
- 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 automatically transfer to GWP and scattering results in thanks to the pseudo-conformal transformation.
- Solitons are -unstable.
