Asymptotic solutions of nonlinear wave equations using the methods of averaging and two-timing

The method of averaging and the two-time procedure are outlined for a class of hyperbolic second-order partial differential equations with small nonlinearities. It is shown that they both lead to the same integro-differential equation for the lowest-order approximation to the solution. For the special case of the nonlinear wave equation, this lowest-order solution consists of the superposition of two modulated travelling waves, and separate integro-differential equations are derived for the amplitudes of these two waves. As an example, the wave equation with Van der Pol type of nonlinearity is considered.