We analyse the eleven integrable equations of the Rosenau–Hyman (RH) family \(\begin{aligned} D_t(u)+D_x(u^m)+D_x^3(u^n)=0, \quad n,m \in \mathbb {Q} \;. \end{aligned}\) These integrable equations were classified in Euler et al. (Discrete Contin Dyn Syst Ser A 40:529–548, 2020). The \(n=m=-2\) case is one of the integrable instances of the RH family. We consider this specific example, and we examine boundedness of solutions and existence and behaviour of travelling waves. We also compute local and nonlocal symmetries for all the integrable RH equations, showing that these equations have very different structural properties; we exhibit some explicit solutions and, finally, we prove that all integrable RH equations describe one-parameter families of pseudo-spherical surfaces and that therefore they may be amenable of analysis via scattering/inverse scattering.
