Time-dependent exact solutions for Rosenau–Hyman equations with variable coefficients

• Rosenau–Hyman equations with time- and space-dependent coefficients.
• Exact time-dependent solutions for vcK(1, 1), vcK(2, 2), and vcK(3, 3) equations.
• Similarity solution related to rarefaction shock wave phenomena.

In this work we study Rosenau–Hyman-like equations that were obtained by imposing the Lie point symmetry algebra of standard KdV to a general K(m,n)K(m,n) equation with variable coefficients. We present time-dependent exact solutions for suited choices of parameters m and n, including the similarity solution related to rarefaction shock wave phenomena.