A new solution representation for the BBM equation in a quarter plane and the eventual periodicity

TL;DR

This paper introduces a novel solution representation for the BBM equation in a quarter plane to analyze its long-term periodic behavior under periodic boundary conditions or forcing, using an inversion formula and stationary phase method.

Contribution

It derives a new formula for solutions of the BBM equation in a quarter plane and proves the eventual periodicity of solutions with periodic data.

Findings

Established the solution representation formula for the BBM equation.

Proved the eventual periodicity of solutions with periodic boundary conditions.

Combined the new formula with stationary phase method to analyze solution behavior.

Abstract

The initial- and boundary-value problem for the Benjamin-Bona-Mahony (BBM) equation is studied in this paper. The goal is to understand the periodic behavior (termed as eventual periodicity) of its solutions corresponding to periodic boundary condition or periodic forcing. To this aim, we derive a new formula representing solutions of this initial- and boundary-value problem by inverting the operator $\partial_t +\alpha \partial_x -\gamma\partial_{xxt}$ defined in the space-time quarter plane. The eventual periodicity of the linearized BBM equation with periodic boundary data and forcing term is established by combining this new representation formula and the method of stationary phase.