What is the *mathematical* and *physical* meaning of the terms *focusing* and *nonfocusing* when they refer to nonlinear terms in a dispersive equations? 



Focusing and nonfocusing nonlinear terms
The terminology focusing versus defocusing comes from electromagnetic wave propagation in a material with a refractive index $n=\alpha|E|^2$ that depends on the energy density of the electric field (proportional to the field strength squared). The corresponding wave equation for propagation along the $z$-axis is the nonlinear Schrödinger equation,
$$ i\frac{\partial E}{\partial z}+\frac{1}{2}\frac{\partial^2 E}{\partial x^2}+ \alpha|E|^2 E = 0.$$


The width in the transverse $x$-direction of the beam increases with increasing $z$ for $\alpha<0$ (*defocusing*), while it decreases for $\alpha>0$ (*focusing*). Defocusing is the normal behavior, the focusing behavior occurs under special circumstances, see the [Wikipedia article.](https://en.wikipedia.org/wiki/Self-focusing)


