NLS scattering

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)

Energy sub-critical, Mass super-critical case

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 $t^{-d/2}\,$ is integrable). In this case one can even relax the $H^{1}\,$ norm to $H^{s}\,$ for some $s<1\,$ 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.

Mass-critical case

Here $p=1+4/d$.

• One can define wave operators assuming that we impose an $L_{x,t}^{p}\,$ 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).

Mass sub-critical case

Here $p<1+4/d$.

• When $p\leq 1+2/d$, 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

$1+8/({\sqrt {d^{2}+12d+4}}+d-2)\,$;

see NaOz2002 for further discussion.