Put $y=x/\lambda(t)$ and
$$
u(t,x)=\lambda^{-1}v(y)e^{-ib|y|^2/4+i\gamma}.
$$
Differentiate the scaling and quadratic phase. After multiplying the nonlinear Schrödinger equation by $\lambda^3e^{ib|y|^2/4-i\gamma}$, the terms proportional to $y\cdot\nabla v+v$, $|y|^2v$, and $v$ vanish respectively when
$$
\frac{ds}{dt}=\lambda^{-2},
\qquad
b=-\lambda^{-1}\frac{d\lambda}{ds},
\qquad
\frac{db}{ds}+b^2=-4,
\qquad
\frac{d\gamma}{ds}=-\omega.
$$
The remaining expression is
$$
-\Delta v+|y|^2v-\omega v+v^3,
$$
which vanishes by part 4. Hence the <Schrödinger expanding lens ansatz for the mass-critical equation> solves the defocusing cubic equation.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-154/1/5/solution