Trapped focusing cubic ground-state equation ($\Delta P_{\eta }-P_{\eta }-\frac{\eta }{4}|x{|}^2{P}_{\eta }+P_{\eta }^3=0$)

For $\eta >0$, the trapped focusing cubic ground-state equation in two dimensions isA nonzero positive solution follows from minimizing the harmonic-oscillator quadratic form subject to a fixed $L^4$ norm and then rescaling by the positive Lagrange multiplier.

If $P_{\eta }$ solves the trapped focusing cubic ground-state equation, the ansatzsolves the two-dimensional mass-critical focusing nonlinear Schrödinger equation when

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]