Focusing nonlinear Schrodinger equation

In the normalizationthe positive cubic term focuses the field and permits bright solitons on zero background.

For every $\kappa >0$ and $x_0\in \mathbb{R}$,solves $i{\psi }_t+{\psi }_{xx}+2|\psi {|}^2\psi =0$ and decays rapidly as $|x|\to \infty$.