On periodic waves governed by the extended Kortewega€“de Vries equation

摘要

The evolution equation describing the propagation of one-dimensional waves in a microstructured material has the form of an extended Kortewega€“de Vries equation, where the additional term reflects the influence of micrononlinearity. As shown by Janno and Engelbrecht (J. Phys. A: Math. Gen., 2005, 38, 5159a€“5172), solitary waves in a microstructured material become asymmetric if nonlinearities are taken into account in both macro- and microscale. The present paper generalizes previous results to periodic waves which, in the KdV case, have the form of cnoidal waves. It is shown that, due to the nonlinearity in microscale, these waves become inclined in the same manner as solitary waves, while the relations between the period, amplitude, and velocity are not affected.