Rational solutions of fifth-order evolutionary equations for describing waves on water

A method is proposed for obtaining the exact solutions of evolutionary equations in the form of a rational function. Invariant manifolds of the equations are used which have the same form of dependence on the required function and its derivatives as the generalized Riccati equations. Using fifth-order Kawahara and Korteweg–de Vries equations as an example, it is shown that their known particular solutions can be obtained using this method. New solutions of a non-linear fifth-order equation, which is encountered when describing long waves on water, are obtained.