Scattering for the quintic generalized Benjamin-Bona-Mahony equation

TL;DR

This paper proves that small, smooth solutions to the quintic generalized Benjamin-Bona-Mahony equation scatter to linear solutions using the space-time resonance method, despite increased complexity in resonance analysis.

Contribution

It extends the space-time resonance analysis to a higher nonlinearity case, handling intricate resonance structures and null conditions.

Findings

Solutions scatter to linear flow for small initial data

Resonance analysis requires refined geometric and analytical techniques

Higher nonlinearity simplifies some interactions but complicates resonance analysis

Abstract

We consider the quintic generalized Benjamin-Bona-Mahony equation $$ u_t-u_{xxt} + \partial_x\big(u + u^{5}\big)= 0,\qquad (t,x)\in \mathbb{R}_+ \times \mathbb{R}. $$ Using the space-time resonance method, we prove that sufficiently small and smooth solutions scatter to the linear flow. While the higher nonlinearity simplifies the treatment of nonresonant interactions compared to the quartic model in \cite{Morgan}, resonance analysis is more intricate. The resonance analysis occurs in a higher-dimensional geometric setting, and certain null or vanishing conditions present in the quartic case fail at specific resonance points. As a result, refined computations and precise estimates near the resonant set are required to close the bootstrap argument.