Quintic NLS

Quintic NLS on $R$

• 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 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 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.

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.

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.

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.