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

Abstract

[For the former parts see: \textit{H. Engin}, \textit{J. M. Ablowitz}, \textit{A. Askar} and \textit{A. S. Cakmak}, ibid. 14, 223-233 (1979; Zbl 0448.73037) and 235-246 (1979; Zbl 0459.73017).] The paper derives the relevant nonlinear integro-differential evolution equation by the method due to \textit{W. D. Collins} [Q. J. Mech. Appl. Math. 24, 129-153 (1971; Zbl 0214.354)] expanding on a procedure by \textit{J. B. Keller} and \textit{L. Ting} [Commun. Pure Appl. Math. 19, 371-420 (1966; Zbl 0284.35004)]. The quadratically nonlinear case is not a trivial variation over the cubically nonlinear 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.