Wave Modulations in Anharmonic Lattices

Abstract

Wave modulations in one-dimensional anharmonic lattices are studied by the use of a perturbation method established by Taniuti and Yajima. A system of equations to determine the evolution of the slowly varying parts in the lowest order of an asymptotic expansion is derived. One interesting result is that the nonlinearly modulated wave must be accompanied by the other slowly varying wave which tends to stabilize the modulated one. '§ I. Introduction Many physical properties of real crystals are directly related to nonlinear effects produced by anharmonic forces. It is important to examine the properties of wave propagations in anharmonic lattices in connection with the thermal expan­ sion of lattices/l the non-ergodic character of nonlinear lattices2J,SJ and the nonlinear propagation of heat observed by Narayanamulti and Varma.4l For special problems on harmonic lattices two- or three-dimensional ones have been studied but the problems on anharmonic lattices, even in the one-dimensional case, are complicated to be studied. Under certain conditions, however, real crystals can be approximated by one-dimensional lattice models.5l Thus in this paper we will focus our attention on the wave modulations in one-dimensional . anharmonic monatomic lattices. Recently, Tappert and Varma5l considered this problem in a continuum limit, assuming that the cubic term in the interaction potential is sufficiently small. In 1970 LowelPl also examined this problem according to Whitham's variational me­ thod6l and showed that for frequencies smaller than a certain critical value the uniform wave is unstable against changes in wavenumber and amplitude. However, he did not take an essential feature acting upon the stabilization of the modulated waves into account. Namely he did not adopt correctly the interactions between an envelope wave and the other slowly varying wave accompanied by the envelope one. In this paper, using the perturbation method established by Taniuti and Yajima/l the wave modulations in anharmonic lattices are investigated. We per­ form the perturbation with due regard to the above effect without the continuum approximation for carrier waves, and obtain the following results: The modulated wave must be accompanied by the other slowly varying wave with the same order as the modulated one, and the both are simultaneously determined with a coupled