Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 154 / 1 / 5 / Solution

Put $y=x/\lambda (t)$ andDifferentiate the scaling and quadratic phase. After multiplying the nonlinear Schrödinger equation by ${\lambda }^3{e}^{ib|y{|}^2/4-i\gamma }$, the terms proportional to $y\cdot \nabla v+v$, $|y{|}^{2}v$, and $v$ vanish respectively whenThe remaining expression iswhich vanishes by part 4. Hence the Schrödinger expanding lens ansatz for the mass-critical equation solves the defocusing cubic equation.