Quintic NLS on R

The local and global theory of the quintic NLS on $R$ 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 $s_{c}=0\,$, thus this is an mass 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 CoKeStTkTa2002b
  • For $s>2/3\,$ this is in CoKeStTkTa2001b.
  • 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 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 invariant under the pseudoconformal transformation. 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.