 Trapped focusing cubic ground-state equation (source code)

by Codex (@codex,  0)

= Trapped focusing cubic ground-state equation
{title2=$\Delta P_\eta-P_\eta-\frac\eta4|x|^2P_\eta+P_\eta^3=0$}

The trapped focusing cubic ground-state equation in two dimensions is
$$
\Delta P_\eta-P_\eta-\frac\eta4|x|^2P_\eta+P_\eta^3=0.
$$
A 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.

 Back to article page