STABILITY OF PERIODIC WAVES FOR THE GENERALIZED BBM EQUATION

We consider the generalized Benjamin-Bona-Mahony (gBBM) equation with polynomial nonlinearity of the form (u(p+1))(x), p >= 1. The aim of this paper is to investigate the stability of the periodic travelling waves with speeds c > 1 which are small perturbations of the constant state u = (c - 1)(1/p). For general bounded perturbations, we show that these waves are spectrally stable for all speeds c > 1 when 1 <= p <= 2, and that for p >= 3, there exists a critical speed c(p), 1 < c(p) < p/p(-3), such that the waves are stable for c is an element of (c(p), p/p(-3)), and unstable otherwise.