I'm searching for a curve in a plane that has a specific curvature, of the form $$ \kappa(s) = A \ \text{sech}(Bs) $$ where $s$ is the arc length parameter, and $A$, $B$ constants. I'm not sure if it is reasonable to try to solve the Frenet-Serret equations analytically, as I have only elementary knowledge of the types of curves in the plane. Is it?
As far as I know, none of the conic curves provides such curvature.
The problem originates from the fact that it is possible to write the motion of a quantum mechanical object in one-dimension by encoding the curvature as a potential. Turns out that the potential is proportional to the curvature squared
$$
V(s)\propto \kappa^2(s)
$$
and a potential of the form $\text{sech}^2(s)$ has very interesting scattering properties.