Schrodinger:quartic NLS: Difference between revisions

From DispersiveWiki
Jump to navigationJump to search
No edit summary
 
No edit summary
Line 1: Line 1:
====Quartic NLS on <math>R</math>====
====Quartic NLS on <math>R</math>====



Revision as of 19:42, 28 July 2006

Quartic NLS on

  • Scaling is sc = -1/6.
  • For any quartic non-linearity one can obtain LWP for s ³ 0 references:CaWe1990 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

  • For any quartic non-linearity one has LWP for s>0 references:Bo1993 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) references:Bo1995c Bo1995c. Indeed one has an invariant measure.

Quartic NLS on

  • Scaling is sc = 1/3.
  • For any quartic non-linearity one can obtain LWP for s ³ sc references:CaWe1990 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 references:Ka1986 Ka1986.