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I'm studying the Nonlinear 1d Schrodinger equation

$$i\psi _t + \psi '' + |\psi |^{2p} \psi - \epsilon |\psi | ^{2q} \psi = 0\, , \quad t>0, x\in \mathbb{R}\, ,$$

and specifically, its solitary solutions, i.e. $\psi (t,x) = e^{i\omega t}R(x) $, which then yields the following ODE:

$$-\omega R + R'' + R^{2p+1} - \epsilon R^{2q+1} = 0 \, . $$

I know that with $p=1$,$q=2$, this is a model for intense laser beams propogation in a 1d waveguide nonlinear Kerr medium with some degree of saturation.

My Question: Does any of these two equations has any physical significance for value of $p$ and $q$ other then $(1,2)$?

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  • $\begingroup$ For $p = q = 1/2$, the equation looks similar to a modified KdV equation, but this fails to satisfy one of your constraints. I know I have seen an equation similar to the one where $p = q = 1$ as well, but again, this fails the $q > p$ constraint. $\endgroup$ – honeste_vivere Jul 25 '16 at 12:56
  • $\begingroup$ @honeste_vivere I've lifted the constraint. Can you say more about it? $\endgroup$ – Amir Sagiv Jul 25 '16 at 16:16
  • $\begingroup$ All the equations to which I refer are for nonlinear waves (specifically solitons and cnoidal waves), but some have slightly different coefficients. This probably should not be too surprising, as the nonlinear Schrodinger equation has been used to describe rogue waves in the Earth's oceans. The primary physical difference is that KdV-like solutions have phase speed dependences on spatial width and amplitude but I think the NLSE does not share this trait (better check to be sure). $\endgroup$ – honeste_vivere Jul 26 '16 at 12:24

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