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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.

A concrete Schrödinger expanding lens ansatz for the mass-critical equation starts with a finite-energy solution $v\in \Sigma$ of the trapped defocusing stationary equationSet $\lambda (t)=\sqrt{1+t^2}$. Thensolves the Defocusing cubic nonlinear Schrödinger equation in two dimensions. Direct substitution gives the scale/chirp conditions $b=-t=-\lambda {\lambda }_t$, ${\lambda }^2{b}_t+b^2=-1$, and phase rate ${\gamma }_t=-\mu /{\lambda }^2$. The displayed stationary equation supplies exactly the remaining spatial terms.^[ref]

The harmonic-oscillator energy space assumption controls both the ordinary gradient and the quadratic-phase gradient. Moreover,by the substitution $x=\lambda y$ and ${\int }_{\mathbb{R}}(1+t^2{)}^{-1}dt=\pi$. This gives an explicit free solution to which scattering from a finite Strichartz norm applies.