NLS scattering

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Once one has global well-posedness of an equation such as NLS, one can ask for scattering properties. Two particular properties of interest are

Asymptotic completeness: Given any initial data in a certain data class, the (global) solution asymptotically converges (in the topology of that class) to a linear solution in that class.
Existence of wave operators: Given a linear solution in a certain data class, there exists a global solution which asymptotically converges to that solution in the topology of that class.

A standard reference is Sr1989.

The scattering behavior depends heavily on the criticality of the exponent, the sign of the nonlinearity, and the size of the data.

Energy-critical case

Here $d \geq 3$ and $p = 1 + 4 / (d - 2)$ .

Scattering in the energy class is now known for large-energy and defocusing nonlinearity in all dimensions three and higher (Visan, Visan-Ryckman, CKSTT)

Here $p > 1 + 4 / d$ . If $d \geq 3$ , we also require $p < 1 + 4 / (d - 2)$ .

Scattering in the energy class for small energy (with either focusing or defocusing nonlinearity) was achieved in Sr1981, Sr1981b.
Scattering in the conformal class $H^1 \cap L^2(|x|^2 dx)$ , large data, defocusing nonlinearity and all dimensions can be achieved using the pseudo-conformal conservation law and Morawetz identities LnSr1978.
Scattering for large energy and defocusing nonlinearity is in GiVl1985 (see also Bo1998b, Na1999c) by exploiting Morawetz identities, approximate finite speed of propagation, and strong decay estimates (the decay of is integrable). In this case one can even relax the norm to for some CoKeStTkTa-p8. For large energy and focusing nonlinearity there is of course blowup.
- For $d=1,2\,$ one can also remove the $L^{2}(|x|^2 dx)\,$ assumption Na1999c by finding a variant of the Morawetz identity for low dimensions, together with Bourgain's induction on energy argument.

Here $p = 1 + 4 / d$ .

One can define wave operators assuming that we impose an $L^p_{x,t}\,$ integrability condition on the solution or some smallness or localization condition on the data GiVl1979, GiVl1985, Bo1998 (see also Ts1985 for the case of finite pseudoconformal charge).
Scattering is now also known in the spherically symmetric case in dimensions three and higher (Tao-Visan-Zhang).

Here $p < 1 + 4 / d$ .

When , standard wave operators do not exist due to the poor decay in time in the non-linear term Bb1984, Gs1977b, Sr1989.
- However at $p = 1+2/d\,$ one can obtain modified wave operators for data with suitable regularity, decay, and moment conditions Oz1991, GiOz1993, HaNm1998, ShiTon2004, HaNmShiTon2004.
One can construct wave operators on certain spaces related to the pseudo-conformal charge CaWe1992, GiOz1993, GiOzVl1994, Oz1991; see also GiVl1979, Ts1985.
For $H^s\,$ wave operators were also constructed in Na2001. However in order to construct wave operators in spaces such as $L^{2}(|x|^2 dx)\,$ (the space of functions with finite pseudoconformal charge) it is necessary that $p\,$ is larger than or equal to the rather unusual power