Resonant non-linear waves—III. Elastic continuum with quadratic non-linearity

The paper derives the relevant non-linear integro-differential evolution equation by the method due to Collins expanding on a procedure by Keller. The quadratically non-linear case is not a trivial variation over the cubically non-linear case that was presented in preceding papers. As expected a different scaling and ordering of terms is needed and the first order perturbation solution provides no information on resonance. Nevertheless, although obtained by much longer calculations, the final equation for the present case is of identical form, with differences only in numerical coefficients, with the cubic case that was presented and solved earlier.