Bäcklund transformation related to two nonlinear differential equations

Question

I'm looking for a Bäcklund transformation linking the following two nonlinear differential equations for real $t$: $$\dfrac{d^2}{dt^2}f(t)=\cos\left[f(t)\right]$$ $$\dfrac{d^2}{dt^2}g(t)=\sinh\left[g(t)\right]$$ where $f$, $g$ are real-valued $C^2$ functions. How can I find it? Thanks

Answer 1

To relate the solutions of these equations, the following simple transformation can be employed: $$g= i \left(\frac\pi2-f\right),$$ where $i$ stands for the imaginary unit, but I don't see any simple way for finding a transformation that would send real solutions of one equations into real solutions of the other.