I am reading this article and am unclear as to how equation $(3.9)$ is obtained. One is supposed to go from
$H''' = \frac{1-H}{H^3}$
to
$\epsilon = A e^{-\zeta} + B \cos(\sqrt{3} \zeta /2 + \phi) e^{\zeta/2}$
(I assume $\epsilon$ is meant to be $H$ here). $\zeta$ is the position coordinate, so corresponds to $x$. This is done by assuming that $H''' \cong 0$ and hence that $H = 1 + \epsilon$, then the equation is linearised and solved. By linearised, is it meant substitute in $1+ \epsilon$, and then throw away all the terms which are higher order in $\epsilon$? How does one end up with $H$ if $H$ has been changed for $1 + \epsilon$.
If there is some reference in lubrication theory where this is explained, that might be helpful as well.