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.

| cite | improve this answer | |
  • $\begingroup$ Thank you. That's very interesting. Do you have a reference for this topic (from the mathematical point of view)? $\endgroup$ – Kei May 23 '19 at 17:38
  • $\begingroup$ Weinstein's review seems a useful introduction. $\endgroup$ – Carlo Beenakker May 23 '19 at 17:48
  • $\begingroup$ Thank you very much. $\endgroup$ – Kei May 24 '19 at 20:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.