Quartic NLS

Quartic NLS on $R$

• Scaling is $s_{c}=-1/6\,$.
• For any quartic non-linearity one can obtain LWP for $s\geq 0\,$ 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 ${\underline {uuuu}}\,$ 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 $u^{4}\,$, $uuu{\underline {u}}\,$, 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 $s\geq 0\,$ 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.

Quartic NLS on $T$

• 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.

Quartic NLS on $R^{2}$

• Scaling is $s_{c}=1/3\,.$
• For any quartic non-linearity one can obtain LWP for $s\geq s_{c}\,$ 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 $s\geq 1\,$ 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.