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

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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.

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  • $\begingroup$ Thank you. That's very interesting. Do you have a reference for this topic (from the mathematical point of view)? $\endgroup$
    – Kei
    May 23, 2019 at 17:38
  • $\begingroup$ Weinstein's review seems a useful introduction. $\endgroup$
    – Carlo Beenakker
    May 23, 2019 at 17:48
  • $\begingroup$ Thank you very much. $\endgroup$
    – Kei
    May 24, 2019 at 20:19

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