# Schrodinger:quartic NLS

#### Quartic NLS on ${\displaystyle R}$

• Scaling is sc = -1/6.
• For any quartic non-linearity one can obtain LWP for s ³ 0 CaWe1990
• Below L^2 we have ill-posedness by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.
• If the quartic non-linearity is of u u u u type then one can obtain LWP for s > -1/6. For |u|4 one has LWP for s > -1/8, while for the other three types u4, u u u u, or u uuu one has LWP for s > -1/6 references#Gr-p2 Gr-p2.
• In the Hamiltonian case (a non-linearity of type |u|^3 u) we have GWP for s ³ 0 by L2 conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.

#### Quartic NLS on ${\displaystyle T}$

• For any quartic non-linearity one has LWP for s>0 Bo1993.
• If the quartic non-linearity is of u u u u type then one can obtain LWP for s > -1/6 references#Gr-p2 Gr-p2.
• If the nonlinearity is of |u|3 u type one has GWP for random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) Bo1995c. Indeed one has an invariant measure.

#### Quartic NLS on ${\displaystyle R^{2}}$

• Scaling is sc = 1/3.
• For any quartic non-linearity one can obtain LWP for s ³ sc CaWe1990.
• 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.
• In the Hamiltonian case (a non-linearity of type |u|^3 u) we have GWP for s ³ 1 Ka1986.
• This has been improved to s > 1-e in CoKeStTkTa2003c in the defocusing Hamiltonian case. This result can of course be improved further.
• Scattering in the energy space Na1999c in the defocusing Hamiltonian case.
• One also has GWP and scattering for small H^{1/3} data for any quintic non-linearity.