Focusing Schrödinger lens ansatz
= Focusing Schrödinger lens ansatz
{c}
If $P_\eta$ solves the <trapped focusing cubic ground-state equation>, the ansatz
$$
u(t,x)=\lambda(t)^{-1}P_\eta(x/\lambda(t))
\exp\left(-\frac{i}{4}b(t)|x/\lambda(t)|^2+i\gamma(t)\right)
$$
solves the two-dimensional <mass-critical focusing nonlinear Schrödinger equation> when
$$
b=-\lambda\lambda_t,
\qquad
\lambda^2b_t+b^2=-\eta,
\qquad
\gamma_t=\lambda^{-2}.
$$