The fifth-order partial differential equation for the description of the {α}+{β} Fermi-Pasta-Ulam model

We study a new nonlinear partial differential equation of the fifth order for the description of perturbations in the Fermi-Pasta-Ulam mass chain. This fifth-order equation is an expansion of the Gardner equation for the description of the Fermi--Pasta--Ulam model. We use the potential of interaction between neighbouring masses with both quadratic and cubic terms. The equation is derived using the continuous limit. Unlike the previous works, we take into account higher order terms in the Taylor series expansions. We investigate the equation using the Painlev\'e approach. We show that the equation does not pass the Painlev\'e test and can not be integrated by the inverse scattering transform. We use the logistic function method and the Laurent expansion method to find travelling wave solutions of the fifth--order equation. We use the pseudospectral method for the numerical simulation of wave processes, described by the equation.