Quintic NLS

Quintic NLS on $R$

• 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 KolNewStrQi2000
• Scaling is $s_{c}=0\,$, thus this is an [#L^2-critical_NLS L^2 critical NLS].
• LWP is known for $s\geq 0\,$ CaWe1990, Ts1987.
  • For $s=0\,$ the time of existence depends on the profile of the data as well as the norm.
  • Below $L^{2}\,$ we have ill-posedness by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.
• GWP for $s>4/9\,$ in the defocussing case [Tzi-p]
  • For $s>1/2\,$ this is in CoKeStTkTa-p6
  • For $s>2/3\,$ this is in CoKeStTkTa-p4.
  • For $s>32/33\,$ this is implicit in Tk-p.
  • For $s\geq 1\,$ this follows from LWP and Hamiltonian conservation.
  • One has GWP and scattering for small $L^{2}\,$ data for any quintic non-linearity. The corresponding problem for large $L^{2}\,$ 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 $L^{6}\,$ norm in spacetime.
  • Explicit blowup solutions (with large $L^{2}\,$ norm) are known in the focussing case BirKnPoSvVe1996. The blowup rate in $H^{1}\,$ is $t^{-1}\,$ in these solutions. This is not the optimal blowup rate; in fact an example has been constructed where the blowup rate is $|t|^{-1/2}(loglog|t|)^{1/2}\,$[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 $L^{2}\,$.
    • 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 pseudo-conformally invariant. GWP results in $H^{s}\,$ automatically transfer to GWP and scattering results in $L^{2}(|x|^{s})\,$ thanks to the pseudo-conformal transformation.
• Solitons are $H^{1}\,$-unstable.

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Quintic NLS on $T$

• This equation may be viewed as a simpler version of cubic DNLS, and is always at least as well-behaved.
• Scaling is $s_{c}=0\,$.
• LWP is known for $s>0\,$ Bo1993.
  • For $s<0\,$ the solution map is not uniformly continuous from $C^{k}\,$ to $C^{-k}\,$ for any $k\,$ [CtCoTa-p3].
• GWP is known in the defocusing case for $s>4/9\,$ (De Silva, Pavlovic, Staffilani, Tzirakis)
  • For $s>2/3\,$ this is commented upon in [Bo-p2] and is a minor modification of [CoKeStTkTa-p].
  • For $s\geq 1\,$ one has GWP in the defocusing case, or in the focusing case with small $L^{2}\,$ norm, by Hamiltonian conservation.
    • In the defocusing case one has GWP for random data whose Fourier coefficients decay like $1/|k|\,$ (times a Gaussian random variable) 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 $L^{2}\,$ norm is sufficiently small.

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Quintic NLS on $R^{2}$

• Scaling is $s_{c}=1/2\,$.
• LWP is known for $s\geq 1/2\,$ CaWe1990.
  • For $s=1/2\,$ the time of existence depends on the profile of the data as well as the norm.
  • 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.
• GWP for $s\geq 1\,$ by Hamiltonian conservation.
  • This has been improved to $s>1-e\,$- in CoKeStTkTa2003b. This result can of course be improved further.
  • Scattering in the energy space Na1999c
  • One also has GWP and scattering for small $H^{1/2}\,$ data for any quintic non-linearity.

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Quintic NLS on $R^{3}$

• Scaling is $s_{c}=1\,$.
• LWP is known for $s\geq 1\,$ CaWe1990.
  • For $s=1\,$ the time of existence depends on the profile of the data as well as the norm.
  • 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.
• GWP and scattering for $s\geq 1\,$ in the defocusing case [CoKeStTkTa-p]
  • For radial data this is in [Bo-p], Bo1999.
  • Blowup can occur in the focussing case from Glassey's virial identity.

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