The Quasi-Periodic Cauchy Problem for the Generalized Benjamin-Bona-Mahony Equation on the Real Line

TL;DR

This paper establishes existence and uniqueness results for the generalized Benjamin-Bona-Mahony equation with quasi-periodic initial data on the real line, extending classical solutions to broader initial conditions.

Contribution

It provides a new existence and uniqueness theorem for the gBBM equation with quasi-periodic data, using combinatorial analysis methods and covering both polynomial and exponential decay cases.

Findings

Existence and uniqueness for polynomial decay initial data.

Extension to exponential decay initial data.

Cauchy-Kovalevskaya type theorem for gBBM with quasi-periodic data.

Abstract

This paper studies the existence and uniqueness problem for the generalized Benjamin-Bona-Mahony (gBBM) equation with quasi-periodic initial data on the real line. We obtain an existence and uniqueness result in the classical sense with arbitrary time horizon under the assumption of polynomially decaying initial Fourier data by using the combinatorial analysis method developed in earlier papers by Christ, Damanik-Goldstein, and the present authors. Our result is valid for exponentially decaying initial Fourier data and hence can be viewed as a Cauchy-Kovalevskaya theorem for the gBBM equation with quasi-periodic initial data.