= Explicit expanding focusing Schrödinger lens
= Explicit lens parameters
{synonym}
Let $\eta>0$ and let $P_\eta\in L^4(\mathbb R^2)$ solve the <trapped focusing cubic ground-state equation>. In the <Focusing Schrödinger lens ansatz>, the initial conditions $\lambda(-1)=\sqrt{1+\eta}$ and $b(-1)=1$ give
$$
\lambda(t)=\sqrt{t^2+\eta},\qquad b(t)=-t,\qquad
\gamma(t)-\gamma(-1)=\int_{-1}^t\frac{d\tau}{\tau^2+\eta}.
$$
Indeed $b=-\lambda\lambda_t$ and $\lambda^2b_t+b^2=-\eta$. The positive $\eta$ keeps the scale nonzero for every real time.
A <change of variables> $y=x/\lambda(t)$ proves the critical spacetime bound exactly:
$$
\int_{\mathbb R}\int_{\mathbb R^2}|u(t,x)|^4\,dx\,dt
=\|P_\eta\|_4^4\int_{\mathbb R}\frac{dt}{t^2+\eta}
=\frac\pi{\sqrt\eta}\|P_\eta\|_4^4.
$$
Thus the solution expands at large times and has finite $L^4$ spacetime norm, rather than merely pointwise decay. This is the norm used by <scattering from a finite Strichartz norm>.<Critical nonlinear Schrödinger equations with and without harmonic potential by Rémi Carles>[Equations (1.4)–(1.5), lens-transform mechanism; the displayed parameter and norm calculations use the wiki coefficient convention]{ref}
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