Hyperbolicity study of the modulation equations for the Benjamin–Bona–Mahony equation

Abstract

A complete study of the modulation equations for the Benjamin-Bona-Mahony equation is performed. In particular, the boundary between the hyperbolic and elliptic regions of the modulation equations is found. When the wave amplitude is small, this boundary is approximately defined by 𝑘 = √ 3, where 𝑘 is the wave number.This particular value corresponds to the inflection point of the linear dispersion relation for the BBM equation. Numerical results are presented showing the appearance of the Benjamin-Feir instability when the periodic solutions are inside the ellipticity region.