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We study a new nonlinear partial differential equation of the fifth order forthe description of perturbations in the Fermi-Pasta-Ulam mass chain. Thisfifth-order equation is an expansion of the Gardner equation for thedescription of the Fermi--Pasta--Ulam model. We use the potential ofinteraction 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. Weinvestigate the equation using the Painlev\'e approach. We show that theequation does not pass the Painlev\'e test and can not be integrated by theinverse scattering transform. We use the logistic function method and theLaurent expansion method to find travelling wave solutions of the fifth--orderequation. We use the pseudospectral method for the numerical simulation of waveprocesses, described by the equation.

The fifth-order partial differential equation for the description of the {\alpha}+{\beta} Fermi-Pasta-Ulam model