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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

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  • $\begingroup$ @LoïcTeyssier: Sorry. I corrected the typos $\endgroup$ – Riccardo.Alestra Jul 8 '14 at 11:52
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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.

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