From the Fermi–Pasta–Ulam Model to Higher-Order Nonlinear Evolution Equations

We consider generalizations of the Korteweg–de Vries equation of the fifth and seventh order obtained from the Fermi–Pasta–Ulam problem. Analytical properties of the equation are investigated taking into account the Painlevé test. It is shown that the equations of the fifth and seventh order do not have the Painlevé property. We demonstrate that there are expansions of the solution in the Laurent series and as a consequence we can find exact solutions of the equations. Solitary wave and elliptic solutions of the fifth and seventh order equations are presented.