# Quartic NLS

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

• Scaling is ${\displaystyle s_{c}=-1/6\,}$.
• For any quartic non-linearity one can obtain LWP for ${\displaystyle s\geq 0\,}$ CaWe1990
• Below ${\displaystyle L^{2}\,}$ 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 ${\displaystyle {\underline {uuuu}}\,}$ type then one can obtain LWP for ${\displaystyle s>-1/6\,.}$ For ${\displaystyle |u|^{4}\,}$ one has LWP for ${\displaystyle s>-1/8\,}$, while for the other three types ${\displaystyle u^{4}\,}$, ${\displaystyle uuu{\underline {u}}\,}$, or ${\displaystyle u{\underline {uuu}}\,}$ one has LWP for ${\displaystyle s>-1/6\,}$ Gr-p2.
• In the Hamiltonian case (a non-linearity of type ${\displaystyle |u|^{3}u\,}$) we have GWP for ${\displaystyle s\geq 0\,}$ by ${\displaystyle L^{2}\,}$ 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 ${\displaystyle s>0\,}$ Bo1993.
• If the quartic non-linearity is of ${\displaystyle {\underline {uuuu}}\,}$ type then one can obtain LWP for ${\displaystyle s>-1/6\,}$, Gr-p2.
• If the nonlinearity is of ${\displaystyle |u|^{3}u\,}$ type one has GWP for random data whose Fourier coefficients decay like ${\displaystyle 1/|k|\,}$ (times a Gaussian random variable) Bo1995c. Indeed one has an invariant measure.

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

• Scaling is ${\displaystyle s_{c}=1/3\,.}$
• For any quartic non-linearity one can obtain LWP for ${\displaystyle s\geq s_{c}\,}$ CaWe1990.
• For ${\displaystyle s we have ill-posedness, indeed the ${\displaystyle H^{s}\,}$ 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 ${\displaystyle |u|^{3}u\,}$) we have GWP for ${\displaystyle s\geq 1\,}$ Ka1986.
• This has been improved to ${\displaystyle s>1-e\,}$ 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 ${\displaystyle H^{1/3}\,}$ data for any quintic non-linearity.