# Focusing and nonfocusing nonlinear terms

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

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.