Quintic NLS on T

From DispersiveWiki

The theory of the quintic NLS on the circle is as follows.

  • This equation may be viewed as a simpler version of cubic DNLS, and is always at least as well-behaved.
  • Scaling is .
  • LWP is known for Bo1993.
    • For the solution map is not uniformly continuous from to for any CtCoTa-p3.
  • GWP is known in the defocusing case for (De Silva, Pavlovic, Staffilani, Tzirakis)
    • For this is commented upon in Bo-p2 and is a minor modification of CoKeStTkTa-p.
    • For one has GWP in the defocusing case, or in the focusing case with small norm, by Hamiltonian conservation.
      • In the defocusing case one has GWP for random data whose Fourier coefficients decay like (times a Gaussian random variable) Bo1995c; this is roughly of the regularity of . Indeed one has an invariant measure. In the focusing case the same result holds assuming the norm is sufficiently small.