Painlevé property and exact solutions for a nonlinear wave equation with generalized power-law nonlinearities

By employing a variety of techniques, we investigate several classes of solutions of a family of nonlinear partial differential equations (NLPDEs) with generalized nonlinearities, special cases of which include the Klein–Gordon equation, the Landau–Ginzburg–Higgs equation, the φ 4 and φ 6 equations, the Rayleigh wave equation. The Painlevé property for our class of equations is studied first, showing that there are integrable families of such equations satisfying the strong Painlevé property (under a traveling wave assumption). From the truncated Laurent expansions, we introduce the auto-Bäcklund transformation for the two families shown to admit the strong Painlevé property. A multi-parameter family of exact solutions is then constructed from these auto-Bäcklund transformations for each of the cases, leading to travelling wave solutions. From here, assuming only travelling wave solutions, we then discuss more general methods of obtaining travelling wave solutions for those cases which do not satisfy the strong Painlevé property. Such solutions constitute rare exact solutions to a complicated nonlinear partial differential equation.