Quartic NLS

Scaling is $s_c = -1/6\,$ .
For any quartic non-linearity one can obtain LWP for 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 $\underline{uuuu}\,$ 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 $u^4\,$ , $u u u \underline{u}\,$ , or $u \underline{uuu}\,$ one has LWP for $s > -1/6\,$ Gr-p2.
In the Hamiltonian case (a non-linearity of type $|u|^3 u\,$ ) we have GWP for $s \ge 0\,$ by $L^2\,$ conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.

For any quartic non-linearity one has LWP for $s>0\,$ Bo1993.
If the quartic non-linearity is of $\underline{uuuu}\,$ type then one can obtain LWP for $s > -1/6\,$ , 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.

Scaling is $s_c = 1/3\,.$
For any quartic non-linearity one can obtain LWP for 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 ) we have GWP for 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.