I continue the study of gradient differential equations, where bell-shaped functions are used as a function, for example:
$\frac{dx}{dt}=\frac{d}{dx}(sech(x)^2)$
The solution of which is:
$x(t) = \sinh ^{-1}\left(\sqrt{LambertW\left(e^{c_1-4 t}\right)}\right)$
where $c_1$ - arbitrary constant.
https://en.wikipedia.org/wiki/Bell_shaped_function
Expansion of nonlinear functions with damping properties in exponential series
Let's assume that $c_1 = 20$.
It can be seen that the solution of such differential equations shows a transition from the initial state to the maximum / minimum point. This transient process is described by a rather complex nonlinear function. I conclude that either an exponential process is at the heart of such a function, over which a nonlinear transformation is performed, or such a transient process is the sum of an exponential process and a wave similar to a lognormal distribution.
I am interested in the following questions:
Are there more or less developed functions of quasi-exponential processes?
Are there more flexible analogs of the lognormal distribution, allowing more fine tuning of its shape, and over which the Laplace transform can be easily performed.