On periodic waves governed by the extended Korteweg–de Vries equation

Abstract

Janno and Engelbrecht [1] have shown that the propagation of nonlinear one-dimensional waves in microstructured solids leads to a generalized Korteweg-deVries equation. After suitable transformation of the variables, the evolution equation of the waves can be written in the form [2] yt + y 2 � x + yxxx + 3 y 2 � xx = 0. (1) The additional term involving the small parameter reflects the nonlinearity in the microscale. Solutions representing asymmetric solitary waves have been analyzed both qualitatively and numerically in [1]. An approximate solution in analytic form has been given in [2]. Solitary waves can be considered as the long-wave limit of periodic solutions which, in the KdV case, have the form of cnoidal waves. The present paper is devoted to periodic solutions y = q(x ct) of the extended KdV equation (1), which emerge from the cnoidal waves for > 0.