I'm trying to understand this article. I think he has missing terms in his equations, and I can't understand how he derived equations 8-10. The math should be straight forward, and this make everything even more annoying. The paper begins with equation
$\frac{d^2 \psi}{dr^2} + \frac{2}{r} \frac{d \psi}{dr} = \kappa^2 \sinh \left ( \psi \right )~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1)$
$\psi$ is an unknown function of $r$, $\kappa$ is some constant, as well as $a$, and the conditions are set so $\frac{1}{\kappa a} \ll 1$
Now, the paper wishes to solve equation (1) next to the surface ($r=0$). The author expands $\psi$ to a power series in what looks like perturbation theory.
$\psi = \psi^{(0)} + \frac{1}{\kappa a} \psi^{(1)} + \frac{1}{\left ( \kappa a \right )^2} \psi ^{(2)} + \ldots ~~~~~~~~\ \ \ (2)$
and he transform the coordinates
$r = a + x/k ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(3)$
to applying (2) and (3) to (1) I first need to apply the chain rules
$\frac{d \psi}{dr} = \kappa \frac{d \psi}{dx}, \frac{d^2 \psi}{dr^2} = \kappa^2 \left ( \frac{d^2 \psi}{dx^2} + \frac{d \psi}{dx} \right)$
Which leads to
$\kappa ^2 \left( \frac{d^2 \psi}{dx^2} + \frac{d \psi}{dx} \right) +\frac{2 \kappa }{a + x} \frac{d \psi}{dx} = \kappa^2 \sinh \left ( \psi \right) ~~~~\ \ (4)$
This can be slightly simplified to
$ \left( \frac{d^2 \psi}{dx^2} + \frac{d \psi}{dx} \right) +\frac{2 / \kappa a}{1 + x/\kappa a} \frac{d \psi}{dx} = \sinh \left ( \psi \right) ~~~~\ \ \ \ \ \ \ ~(5)$
However, the article reads
$ \frac{d^2 \psi}{dx^2} +\frac{2 / \kappa a}{1 + x/\kappa a} \frac{d \psi}{dx} = \sinh \left ( \psi \right) ~~~~\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (6)$
Which means that use of the chain rule was faulty, but I can't understand why. Can you explain?
The next problem I have is with equating the terms of the same power in $\kappa a$, I can't see the justification why, except from assuming that $1/\kappa a$ is small enough. Is that the case?
Finally, the article turns the $\sinh$ terms into $\sinh \left ( \psi^{(0)} \right ) $, $\frac{1}{\kappa a} \psi ^{(1)} \cosh \left ( \psi^{(0)} \right ) $, and $\frac{1}{\left(\kappa a \right)^2} \frac{1}{2} \left( \psi ^{(1)}\right)^2 \sinh \left ( \psi^{(0)} \right ) + \frac{1}{\left( \kappa a \right)^2} \psi ^{(2)} \cosh \left ( \psi^{(0)} \right )$. This looks some sort of expansion
$f(x_0) + f'(x_0) x_1 + \frac{1}{2} f''(x_0) x_2$
Specifically, this reminds me of the perturbation in equation 2, but I'm not sure this is the case. Is it the case? Why is it justified here?