# Difference between revisions of "Quintic NLS"

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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. | ** 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] | * GWP for s>4/9 in the defocussing case [Tzi-p] | ||

− | ** For s>1/2 this is in [[ | + | ** For s>1/2 this is in [[Bibliography#CoKeStTkTa-p6 |CoKeStTkTa-p6]] |

− | ** For s>2/3 this is in [[ | + | ** For s>2/3 this is in [[Bibliography#CoKeStTkTa-p4 |CoKeStTkTa-p4]]. |

− | ** For s > 32/33 this is implicit in [[ | + | ** For s > 32/33 this is implicit in [[Bibliography#Tk-p |Tk-p]]. |

** For s<font face="Symbol">³</font> 1 this follows from LWP and Hamiltonian conservation. | ** For s<font face="Symbol">³</font> 1 this follows from LWP and Hamiltonian conservation. | ||

** 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. | ** 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. |

## Revision as of 16:46, 31 July 2006

#### Quintic NLS on

- 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 ³ 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 s³ 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} (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.- 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.

#### Quintic NLS on

- 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 >= 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.

- 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

#### Quintic NLS on

- Scaling is s
_{c}= 1/2. - LWP is known for s ³ 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 ³ 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

- Scaling is s
_{c}= 1. - LWP is known for s ³ 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 s³1 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.