Cubic NLS on R
The theory of the cubic NLS on one dimension is as follows.
- LWP for Ts1987, CaWe1990 (see also GiVl1985).
- This is sharp for reasons of Gallilean invariance and for soliton solutions in the focussing case KnPoVe2001
- For initial data equal to a delta function there are serious problems with existence and uniqueness KnPoVe2001.
- However, there exist Gallilean invariant spaces which scale below for which one has LWP. They are defined in terms of the Fourier transform VaVe2001. For instance one has LWP for data whose Fourier transform decays like . Ideally one would like to replace this with
- GWP for thanks to conservation.
- GWP can be pushed below to certain of the Gallilean spaces in VaVe2001. For instance one has GWP when the Fourier transform of the data decays like . Ideally one would like to replace this with 0 − .
- If the cubic non-linearity is of or type (as opposed to the usual type) then one can obtain LWP for Gr-p2. If the nonlinearity is of type then one has LWP for Gr-p2.
- Remark: This equation is sometimes known as the Zakharov-Shabat equation and is completely integrable (see e.g. 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 . By translation and phase shift one thus obtains a four-dimensional manifold of ground states (aka solitons) for each energy. This manifold is -stable Ws1985, Ws1986. Below the energy norm orbital stability is not known, however there are polynomial bounds on the instability CoKeStTkTa2003b.
- This equation is related to the evolution of vortex filaments under the localized induction approximation, via the Hasimoto transformation, see e.g. 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 Oz1991, see also CtCoTa-p. For small, decaying data one also has asymptotic completeness HaNm1998.
- For large Schwartz data, these asymptotics can be obtained by inverse scattering methods ZkMan1976, SeAb1976, No1980, DfZx1994
- For large real analytic data, these asymptotics were obtained in GiVl2001
- Refinements to the convergence and regularity of the modified wave operators was obtained in Car2001.