Symmetry analysis of the high-order equations for the description of the Fermi – Pasta – Ulam problem

Recently some new nonlinear equations for the description of the Fermi – Pasta – Ulam problem have been derived. The main aim of this work is to use the symmetry test to investigate these equations. We consider equations for the description of the α and α + β Fermi – Pasta – Ulam model. We find the infinitesimal operators and Lie groups, admitted by the equations. Using the groups we find the self-similar variables as well as the reductions to the ordinary differential equations. Some exact solutions are also constructed. (paper)