Quartic NLS

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Quartic NLS on

  • Scaling is .
  • For any quartic non-linearity one can obtain LWP for CaWe1990
    • Below 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 type then one can obtain LWP for For one has LWP for , while for the other three types , , or one has LWP for Gr-p2.
  • In the Hamiltonian case (a non-linearity of type ) we have GWP for by 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

  • For any quartic non-linearity one has LWP for Bo1993.
  • If the quartic non-linearity is of type then one can obtain LWP for , Gr-p2.
  • If the nonlinearity is of type one has GWP for random data whose Fourier coefficients decay like (times a Gaussian random variable) Bo1995c. Indeed one has an invariant measure.

Quartic NLS on

  • Scaling is
  • For any quartic non-linearity one can obtain LWP for CaWe1990.
    • For we have ill-posedness, indeed the 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 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 data for any quintic non-linearity.