UC Santa Barbara

Wave propagation phenomena often involve the study of dispersive partial differential equations, which is defined by the linear terms of the system considered. An interesting case arises when a model also incorporates nonlocal operators. In this work, we focus on the Benjamin-Bona-Mahony (BBM) equation and the b-family of peakon equations, which contain the nonlocal operator (1−∂ 2 x ) −1 , with the goal of studying unique continuation properties, smoothing properties, and local energy decay properties of the respective aforementioned systems.Such properties can often be derived from dispersive estimates. However, the BBM equation has very weak dispersion relation and the b-family equations are not dispersive, so such techniques are unavailable. We instead derive the desired properties via methods that rely heavily upon the nonlinear terms of the respective equations.