The theory of the quintic NLS on is as follows.
- Scaling is .
- LWP is known for CaWe1990.
- For the time of existence depends on the profile of the data as well as the norm.
- For 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.
- GWP for by Hamiltonian conservation.
- This has been improved to in CoKeStTkTa2003b. This result can of course be improved further.
- Scattering in the energy space Na1999c
- One also has GWP and scattering for small data for any quintic non-linearity.