Pseudoconformal Transformation

The pseudoconformal transformation

$u(t,x)\to v(\tau ,y)=|\tau |^{-{\frac {d}{2}}}e^{\frac {i|y|^{2}}{4t}}u(-{\frac {1}{\tau }},{\frac {y}{\tau }})$

maps solutions of the mass critical NLS to solutions. The pseudoconformal image of a soliton is an explicit blowup solution. The pseudoconformal transformation is an isometry on $L_{x}^{2}$ and on $L^{2}$-admissible Strichartz spaces $L_{\tau }^{q}L_{y}^{r}$.