We describe recent work with Jim Colliander, Gigliola Staffilani, Hideo Takaoka, and Terry Tao on the following nonlinear Schr¨odinger equation in three space dimensions,

∂tφ + i∆φ = |φ| 4φ

where we start the evolution from given data φ0(x) at time zero. This equation is “critical” in the sense that the natural scaling of the equation leaves the conserved energy norm of solutions unchanged. (So for example, we can’t reduce to the case of small energy data simply by scaling.) Our result is that if the initial data has finite (and possibly large) energy, then there is a unique solution that exists for all time and asymptotically approaches a linear solution. Our work here starts from the results and methods obtained in the context of radially symmetric solutions by J. Bourgain in JAMS 1999, and also the work for radial data by M. Grillakis in CPAM 2000.