Schrodinger:cubic NLS

Cubic NLS on R

• Scaling is s_c = -1/2.
• LWP for s ³ 0 references:Ts1987 Ts1987, references:CaWe1990 CaWe1990 (see also references:GiVl1985 GiVl1985).
  • This is sharp for reasons of Gallilean invariance and for soliton solutions in the focussing case [KnPoVe-p]
    • The result is also sharp in the defocussing case [CtCoTa-p], due to Gallilean invariance and the asymptotic solutions in references:Oz1991 Oz1991.
    • Below s ³0 the solution map was known to be not C² in references:Bo1993 Bo1993
  • For initial data equal to a delta function there are serious problems with existence and uniqueness [KnPoVe-p].
  • However, there exist Gallilean invariant spaces which scale below L² for which one has LWP. They are defined in terms of the Fourier transform references:VaVe2001 VaVe2001. For instance one has LWP for data whose Fourier transform decays like |x|^{-1/6-}. Ideally one would like to replace this with |x|^{0-}.
• GWP for s ³ 0 thanks to L² conservation
  • GWP can be pushed below to certain of the Gallilean spaces in [VaVe-p]. For instance one has GWP when the Fourier transform of the data decays like |x|^{-5/12-}. Ideally one would like to replace this with 0-.
• If the cubic non-linearity is of u u u or u u u type (as opposed to the usual |u|² u type) then one can obtain LWP for s > -5/12 references#Gr-p2 Gr-p2. If the nonlinearity is of u u u type then one has LWP for s > -2/5 references#Gr-p2 Gr-p2.
• Remark: This equation is sometimes known as the Zakharov-Shabat equation and is completely integrable (see e.g. [[references:AbKauNeSe1974 AbKauNeSe1974]]; all higher order integer Sobolev norms stay bounded. Growth of fractional norms might be interesting, though.
• In the focusing case there are soliton and multisoliton solutions, however the defocusing case does not admit such solutions.
• In the focussing case there is a unique positive radial ground state for each energy E. By translation and phase shift one thus obtains a four-dimensional manifold of ground states (aka solitons) for each energy. This manifold is H¹-stable references:Ws1985 Ws1985, references:Ws1986 Ws1986. Below the energy norm orbital stability is not known, however there are polynomial bounds on the instability references:CoKeStTkTa2003b CoKeStTkTa2003b.
• This equation is related to the evolution of vortex filaments under the localized induction approximation, via the Hasimoto transformation, see e.g. references:Hm1972 Hm1972
• Solutions do not scatter to free Schrodinger solutions. In the focussing case this can be easily seen from the existence of solitons. But even in the defocussing case wave operators do not exist, and must be replaced by modified wave operators references:Oz1991 Oz1991, see also [CtCoTa-p]. For small, decaying data one also has asymptotic completeness references:HaNm1998 HaNm1998.
  • For large Schwartz data, these asymptotics can be obtained by inverse scattering methods references:ZkMan1976 ZkMan1976, references:SeAb1976 SeAb1976, references:No1980 No1980, references:DfZx1994 DfZx1994
  • For large real analytic data, these asymptotics were obtained in references:GiVl2001 GiVl2001
  • Refinements to the convergence and regularity of the modified wave operators was obtained in references:Car2001 Car2001
• On the half line R^+, global well-posedness in H^2 was established in references:CrrBu.1991 CrrBu.1991, references:Bu.1992 Bu.1992
• On the interval, the inverse scattering method was applied to generate solutions in [GriSan-p].

---