 Defocusing cubic nonlinear Schrödinger equation in two dimensions

The two-dimensional defocusing cubic nonlinear Schrödinger equation is mass-critical: the scaling $u(t,x)\mapsto {\lambda }^{-1}u(t/{\lambda }^2,x/\lambda )$ preserves the $L^2$ norm. The defocusing sign makes its nonlinear energy positive.