# Schrodinger:quartic NLS

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Jump to navigationJump to search#### Quartic NLS on

- 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 u^{4}, 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 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

- 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

- 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.