Quartic NLS

Quartic NLS on $R$

• Scaling is s_c = -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 $T$

• 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 $R^{2}$

• Scaling is s_c = 1/3.
• For any quartic non-linearity one can obtain LWP for s ³ 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 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.