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arXiv · 2402.09605

Scattering for the Quartic Generalized Benjamin-Bona-Mahony Equation

A. George Morgan
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Abstract

The generalized Benjamin-Bona-Mahony equation (gBBM) is a model for nonlinear dispersive waves which, in the long-wave limit, 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.

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A. George Morgan. 2024-02-14. Scattering for the Quartic Generalized Benjamin-Bona-Mahony Equation. https://arxiv.org/abs/2402.09605

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