On the averaging method for rod equations with quadratic non-linearity

The averaging method is used to approximate solutions of systems of linearly coupled, (quadratic) non-linear dispersive wave equations, which describe extensional–torsional dynamics of a rod. Existence and uniqueness results are established. Error estimates confirm the asymptotic validity of the approximation method on a long time-scale. The linear couplings between the equations imply that resonance can occur inside a single mode of the solution, but energy can also be transferred to other modes.