While reading some papers about the nonlinear schrodinger equation (NLS) I noticed that the authors sometimes use (for the linear case) $$\partial_zE=\frac{i}{2k_0}\nabla^2E$$ and sometimes $$\partial_tE=\frac{i}{2k_0}\nabla^2E$$ I assume it is possible to rewrite the first equation as $$\partial_zE=\frac{\partial t}{\partial z}\frac{\partial E}{\partial t}$$ and by calculating $\partial_zt$ I just get a prefactor for the conversion.
Is that correct?
Assumed it is, I would calculate $\partial_zt$ as $\frac{n}{c}$, with $n$ the refractive index, and $c$ the speed of light in vacuum, resulting in a constant prefactor (assuming $n$ is constant, too).

  • $\begingroup$ A choice of an evolutional variable ($t$ or $z$) depends, mainly, on a way to form initial (and boundary) conditions (IC). If at time $t = 0$, we create some field distribution in the system, and then observe the dynamics at late times, then, clearly, the evolutional variable is $t$. Another way to generate a wave is to excite a medium at point $z=0$. For example, we input light to a medium at $z = 0$ ($z$ is the longitudinal direction). A device at $z=0$ measures the field for all times $t$. This is our IC, and we observe the field evolution on $z$, while $t$ is the "spatial" variable. $\endgroup$ – user2320292 Dec 27 '19 at 7:13

When we use the NLS for quantum systems such as the Gross-Pitaevskii equation then $t$ is time. When we use the NLS equation for paraxial non-linear optics then $z$ is the distance down the optical axis or the distance down the fibre. It's mathematically the same equation but different applications, so different notation. There is no changing $z$ to $t$ by the chain rule.

Suppose, for example, that the dispersive properties of an optical fibre are such that the associated wavenumber for frequencies near $\omega_0$ can be expanded as $$ k= \Delta k+ k_0 + \beta_1 (\omega-\omega_0)+ \frac 12 \beta_2(\omega-\omega_0)^2+\cdots. $$ Here, $\beta_1$ is the reciprocal of the group velocity, and $\beta_2$ is a parameter called the group velocity dispersion (GVD). The term $\Delta k$ parameterizes the change in refractive index due to non-linear effects. It is proportional to the mean-square of the electric field. Let us write the electric field as $$ E(x,t) = A(x,t) e^{ik_0z -\omega_0t}, $$ where $A(x,t)$ is a slowly varying envelope function. When we transform from Fourier variables to space and time we have
$$ (\omega-\omega_0) \to i\frac{\partial}{\partial t},\quad (k-k_0) \to -i \frac{\partial}{\partial z}, $$ and so the equation determining $A$ becomes $$ -i\frac{\partial A }{\partial z}= i\beta_1 \frac{\partial A }{\partial t} -\frac {\beta_2}{2}\frac{\partial^2 A }{\partial t^2}+ \Delta k A. $$ If we set $\Delta k= \gamma |A^2|$, where $\gamma$ is usually positive, we have $$ i\left(\frac{\partial A }{\partial z}+\beta_1 \frac{\partial A }{\partial t}\right)= \frac {\beta_2}{2}\frac{\partial^2 A }{\partial t^2}- \gamma|A|^2 A. $$ We may get rid of the first-order time derivative by transforming to a frame moving at the group velocity. We do this by setting $$ \tau= t-\beta_1z,\\ \zeta = z $$ and using the chain rule. The equation for $A$ ends up being $$ i\frac{\partial A }{\partial \zeta}= \frac {\beta_2}{2}\frac{\partial^2 A }{\partial \tau^2}- \gamma|A|^2 A. $$ This looks like the usual non-linear Schroedinger equation, but with the role of space and time interchanged! Also, the coefficient of the second derivative has the wrong sign! To make it coincide with the Schoedinger equation with solitons we must
have $\beta_2<0$. When this condition holds, we are said to be in the anomalous dispersion regime --- although this is rather a misnomer since it is the group refractive index, $N_g= c/v_{\rm group}$, that is decreasing with frequency, not the ordinary refractive index. For pure SiO$_2$ glass, $\beta_2$ is negative for wavelengths greater than $1.27\, \mu{\rm m}$. We therefore have anomalous dispersion in the technologically important region near $1.55\, \mu{\rm m}$, where the glass is most transparent. In the anomalous dispersion regime we have solitons with $$ A(\zeta,\tau)=e^{i\alpha|\beta_2|\zeta/2} \sqrt{\frac{\beta_2\alpha}{\gamma}}{\rm sech\,}\sqrt{\alpha}(\tau), $$ leading to
$$ E(z,t) = \sqrt{\frac{\beta_2\alpha}{\gamma}}{\rm sech\,}\sqrt{\alpha}(t-\beta_1 z)e^{i\alpha|\beta_2|z/2}e^{ik_0z-i\omega_0t}. $$ This equation describes a pulse propagating at $\beta_1^{-1}$, which is the group velocity.

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  • $\begingroup$ Assumed I want to couple the equation written using $\partial_z$, i.e. a pulse propagating along a fibre, with another equation using $\partial_t$ (describing the internal processes in the whole fibre), how would I do that in this case (which was the motivation for my question), i.e. bring both equations into the same form? $\endgroup$ – arc_lupus Jul 2 '19 at 17:10
  • $\begingroup$ Ill add a note on this to my answer. $\endgroup$ – mike stone Jul 2 '19 at 19:55

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