According to my textbook the nonlinear Schrödinger equation:(adsbygoogle = window.adsbygoogle || []).push({});

$$\frac{\partial A(z,T)}{\partial z} = -i \frac{\beta_2}{2} \frac{\partial^2A}{\partial T^2} + i \gamma |A|^2 A \ \ (1)$$

can be cast in the form

$$\frac{\partial U(z,\tau)}{\partial z} = -i \frac{sign \beta_2}{2} \frac{1}{L_D} \frac{\partial^2 U}{\partial \tau^2} + i \frac{1}{L_{NL}} |U|^2 U \ \ (2)$$

by normalizing with: ##\tau = \frac{T}{T_0},## and ##A(z,T) = \sqrt{P_0} U(z, \tau).##

But my textbook does not show the steps involved, and I can't arrive at equation(2)when I try to do this myself.

So substituting the two parameters into(1)we get

$$\frac{\partial (\sqrt{P_0} U(z, \tau))}{\partial z} = -i \frac{\beta_2}{2} \frac{\partial^2(\sqrt{P_0} U(z, \tau))}{\partial (\tau T_0)^2} + i \gamma |(\sqrt{P_0} U(z, \tau))|^2 (\sqrt{P_0} U(z, \tau))$$

We know that the dispersion length is given by ##L_D = \frac{T_0^2}{|\beta_2|}## and the the nonlinear length is ##L_{NL} = \frac{1}{\gamma P_0}.## When substituting these two the expression becomes

$$\frac{\partial (\sqrt{P_0} U(z, \tau))}{\partial z} = -i \frac{1}{2} \frac{1}{L_D} \frac{\partial^2(\sqrt{P_0} U(z, \tau))}{\partial \tau^2} + i \frac{1}{L_{NL}} \sqrt{P_0} P_0 U^2 U.$$

So, what can we do about the extra ##\sqrt{P_0}##'s and the extra ##P_0## (peak power)? What is wrong here?

Any explanation is greatly appreciated.

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# A The Nonlinear Schrödinger Equation

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