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| ====Quintic NLS on <math>R</math>====
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| * 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]]
| | The quintic [[NLS]] has been studied [[Quintic NLS on R|on R]], [[Quintic NLS on R|on T]], |
| * Scaling is <math>s_c = 0\,</math>, thus this is an [[mass critical NLS]].
| | [[Quintic NLS on R2|on R^2]], and [[Quintic NLS on R3|on R^3]]. |
| * LWP is known for <math>s \ge 0\,</math> [[CaWe1990]], [[Ts1987]].
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| ** For <math>s=0\,</math> the time of existence depends on the profile of the data as well as the norm.
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| ** Below <math>L^2\,</math> we have ill-posedness by Gallilean invariance considerations in both the focusing [[KnPoVe-p]] and defocusing [[CtCoTa-p2]] cases.
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| * GWP for <math>s>4/9\,</math> in the defocussing case [[Tzi-p]]
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| ** For <math>s>1/2\,</math> this is in [[CoKeStTkTa-p6]]
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| ** For <math>s>2/3\,</math> this is in [[CoKeStTkTa-p4]].
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| ** For <math>s > 32/33\,</math> this is implicit in [[Tk-p]].
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| ** For <math>s\ge 1\,</math> this follows from LWP and Hamiltonian conservation.
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| ** 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.
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| ** 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>.
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| *** 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]].
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| * ''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.
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| * Solitons are <math>H^1\,</math>-unstable.
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| ====Quintic NLS on <math>T</math>====
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| * This equation may be viewed as a simpler version of [[cubic DNLS]], and is always at least as well-behaved.
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| * Scaling is <math>s_c = 0\,</math>.
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| * LWP is known for <math>s > 0\,</math> [[Bo1993]].
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| ** For <math>s < 0\,</math> the solution map is not uniformly continuous from <math>C^k\,</math> to <math>C^{-k}\,</math> for any <math>k\,</math> [[CtCoTa-p3]].
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| * GWP is known in the defocusing case for <math>s > 4/9\,</math> (De Silva, Pavlovic, Staffilani, Tzirakis)
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| ** For <math>s > 2/3\,</math> this is commented upon in [[Bo-p2]] and is a minor modification of [[CoKeStTkTa-p]].
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| ** For <math>s \ge 1\,</math> one has GWP in the defocusing case, or in the focusing case with small <math>L^2\,</math> norm, by Hamiltonian conservation.
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| *** In the defocusing case one has GWP for random data whose Fourier coefficients decay like <math>1/|k|\,</math> (times a Gaussian random variable) [[Bo1995c]]; this is roughly of the regularity of <math>H^{1/2}\,</math>. Indeed one has an invariant measure. In the focusing case the same result holds assuming the <math>L^2\,</math> norm is sufficiently small.
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| ====Quintic NLS on <math>R^2</math>====
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| * Scaling is <math>s_c = 1/2\,</math>.
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| * LWP is known for <math>s \ge 1/2\,</math> [[CaWe1990]].
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| ** For <math>s=1/2\,</math> the time of existence depends on the profile of the data as well as the norm.
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| ** For <math>s<s_c\,</math> we have ill-posedness, indeed the H^s norm can get arbitrarily large arbitrarily quickly [[CtCoTa-p2]]. In the focusing case we have instantaneous blowup from the virial identity and scaling.
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| * GWP for <math>s \ge 1\,</math> by Hamiltonian conservation.
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| ** This has been improved to <math>s > 1-e\,</math>- in [[CoKeStTkTa2003b]]. This result can of course be improved further.
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| ** Scattering in the energy space [[Na1999c]]
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| ** One also has GWP and scattering for small <math>H^{1/2}\,</math> data for any quintic non-linearity.
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| ====Quintic NLS on <math>R^3</math>====
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| * Scaling is <math>s_c = 1\,</math>.
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| * LWP is known for <math>s \ge 1\,</math> [[CaWe1990]].
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| ** For <math>s=1\,</math> the time of existence depends on the profile of the data as well as the norm.
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| ** For <math>s<s_c\,</math> we have ill-posedness, indeed the <math>H^s\,</math> norm can get arbitrarily large arbitrarily quickly [[CtCoTa-p2]]. In the focusing case we have instantaneous blowup from the virial identity and scaling.
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| * GWP and scattering for <math>s\ge 1\,</math> in the defocusing case [[CoKeStTkTa-p]]
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| ** For radial data this is in [[Bo-p]], [[Bo1999]].
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| ** Blowup can occur in the focussing case from Glassey's virial identity.
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| [[Category:Schrodinger]] | | [[Category:Schrodinger]] |
| [[Category:Equations]] | | [[Category:Equations]] |
