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| | | #REDIRECT [[quintic NLS]] |
| ====Quintic NLS on <math>R</math>====
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| * This equation may be viewed as a simpler version of [#dnls-3_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 [[references:KolNewStrQi2000 KolNewStrQi2000]]
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| * Scaling is s<sub>c</sub> = 0, thus this is an [#L^2-critical_NLS L^2 critical NLS].
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| * LWP is known for s <font face="Symbol">³</font> 0 [[references:CaWe1990 CaWe1990]], [[references:Ts1987 Ts1987]].
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| ** For s=0 the time of existence depends on the profile of the data as well as the norm.
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| ** Below L^2 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 s>4/9 in the defocussing case [Tzi-p]
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| ** For s>1/2 this is in [[references:CoKeStTkTa-p6 CoKeStTkTa-p6]]
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| ** For s>2/3 this is in [[references:CoKeStTkTa-p4 CoKeStTkTa-p4]].
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| ** For s > 32/33 this is implicit in [[references:Tk-p Tk-p]].
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| ** For s<font face="Symbol">³</font> 1 this follows from LWP and Hamiltonian conservation.
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| ** One has GWP and scattering for small L<sup>2</sup> data for any quintic non-linearity. The corresponding problem for large L<sup>2</sup> data and defocussing nonlinearity is very interesting, but probably very difficult, perhaps only marginally easier than the corresponding problem for the [#Cubic_NLS_on_R^2 2D cubic NLS]. It would suffice to show that the solution has a bounded L<sup>6</sup> norm in spacetime.
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| ** Explicit blowup solutions (with large L<sup>2</sup> norm) are known in the focussing case [[references:BirKnPoSvVe1996 BirKnPoSvVe1996]]. The blowup rate in H<sup>1</sup> is t<sup>-1</sup> in these solutions. This is not the optimal blowup rate; in fact an example has been constructed where the blowup rate is |t|^{-1/2} (log log|t|)^{1/2}[Per-p]. Furthermore, one always this blowup behavior (or possibly slower, though one must still blow up by at least |t|^{-1/2}) whenever the energy is negative [MeRap-p], [MeRap-p2], and one either assumes that the mass is close to the critical mass or that xu is in L^2.
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| *** One can modify the explicit solutions from [[references:BirKnPoSvVe1996 BirKnPoSvVe1996]] and in fact create solutions which blow up at any collection of specified points in spacetime [[references:BoWg1997 BoWg1997]], [[references:Nw1998 Nw1998]].
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| * ''Remark''<nowiki>: This equation is pseudo-conformally invariant. GWP results in H</nowiki><sup>s</sup> automatically transfer to GWP and scattering results in L<sup>2</sup>(|x|<sup>s</sup>) thanks to the pseudo-conformal transformation.
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| * Solitons are H<sup>1</sup>-unstable.
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| <div class="MsoNormal" style="text-align: center"><center>
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| ----
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| </center></div>
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| [[Category:Equations]]
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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 s<sub>c</sub> = 0.
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| * LWP is known for s > 0 [[references:Bo1993 Bo1993]].
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| ** For s < 0 the solution map is not uniformly continuous from C^k to C^{-k} for any k [CtCoTa-p3].
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| * GWP is known in the defocusing case for s > 4/9 (De Silva, Pavlovic, Staffilani, Tzirakis)
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| ** For s > 2/3 this is commented upon in [Bo-p2] and is a minor modification of [CoKeStTkTa-p].
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| ** For s >= 1 one has GWP in the defocusing case, or in the focusing case with small L^2 norm, by Hamiltonian conservation.
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| *** In the defocusing case one has GWP for random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) [[references:Bo1995c Bo1995c]]; this is roughly of the regularity of H^{1/2}. Indeed one has an invariant measure. In the focusing case the same result holds assuming the L<sup>2</sup> norm is sufficiently small.
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| <div class="MsoNormal" style="text-align: center"><center>
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| ----
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| </center></div>
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| [[Category:Equations]]
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| ====Quintic NLS on <math>R^2</math>====
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| * Scaling is s<sub>c</sub> = 1/2.
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| * LWP is known for s <font face="Symbol">³</font> 1/2 [[references:CaWe1990 CaWe1990]].
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| ** For s=1/2 the time of existence depends on the profile of the data as well as the norm.
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| ** For s<s_c 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 s <font face="Symbol">³</font> 1 by Hamiltonian conservation.
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| ** This has been improved to s > 1-<font face="Symbol">e</font> in [[references:CoKeStTkTa2003b CoKeStTkTa2003b]]. This result can of course be improved further.
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| ** Scattering in the energy space [[references:Na1999c Na1999c]]
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| ** One also has GWP and scattering for small H^{1/2} data for any quintic non-linearity.
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| <div class="MsoNormal" style="text-align: center"><center>
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| ----
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| </center></div>
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| [[Category:Equations]]
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| ====Quintic NLS on <math>R^3</math>====
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| * Scaling is s<sub>c</sub> = 1.
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| * LWP is known for s <font face="Symbol">³</font> 1 [[references:CaWe1990 CaWe1990]].
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| ** For s=1 the time of existence depends on the profile of the data as well as the norm.
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| ** For s<s_c 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 and scattering for s<font face="Symbol">³</font>1 in the defocusing case [CoKeStTkTa-p]
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| ** For radial data this is in [Bo-p], [[references:Bo1999 Bo1999]].
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| ** Blowup can occur in the focussing case from Glassey's virial identity.
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| <div class="MsoNormal" style="text-align: center"><center>
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| ----
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| </center></div>
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| [[Category:Equations]]
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