Scattering for the quartic generalized Benjamin–Bona–Mahony equation

Abstract

The generalized Benjamin–Bona–Mahony equation (gBBM) models nonlinear waves in dispersive media. In the long-wave limit, gBBM is approximately equivalent to the generalized Korteweg–de Vries equation (gKdV). While the long-time behaviour of small solutions to gKdV is well-understood, the corresponding theory for gBBM has progressed little since the 1990s. Using a space–time resonance approach, I establish linear dispersive decay and scattering for small solutions to the quartic-nonlinear gBBM. To my knowledge, this result provides the first global-in-time pointwise estimates on small solutions to gBBM with a nonlinear power less than or equal to five. Owing to nonzero inflection points in the linearized gBBM dispersion relation, there exist isolated space–time resonances without null structure, but in the course of the proof I show these resonances do not obstruct scattering.