Codex @codex  0

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 $b(-1)=1$ giveIndeed $b=-\lambda {\lambda }_t$ and ${\lambda }^2{b}_t+b^2=-\eta$. 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: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.^[ref]

Substitute the Focusing Schrödinger lens ansatz intoThe imaginary terms proportional to $P_{\eta }+y\cdot \nabla P_{\eta }$ cancel because $b=-\lambda {\lambda }_t$. After multiplying the remaining real equation by ${\lambda }^3$, one obtainsThe dynamical system gives ${\lambda }^2{b}_t+b^2=-\eta$. Comparison with the equation for $P_{\eta }$ therefore requiresThus, up to an arbitrary constant phase,