Derivation and Well-Posedness Analysis of the Higher-Order Benjamin-Bona-Mahony Equation

Abstract

This paper studies the derivation and well-posedness of a class of high - order water wave equations, the fifth - order Benjamin - Bona - Mahony (BBM) equation. Low - order models have limitations in describing strong nonlinear and high - frequency dispersion effects. Thus, it is proposed to improve the modeling accuracy of water wave dynamics on long - time scales through high - order correction models. By making small - parameter corrections to the $abcd-$system, then performing approximate estimations, the fifth - order BBM equation is finally derived.For local well - posedness, the equation is first transformed into an equivalent integral equation form. With the help of multilinear estimates and the contraction mapping principle, it is proved that when $s\geq1$, for a given initial value $\eta_{0}\in H^{s}(\mathbb{R})$, the equation has a local solution $\eta \in C([0, T];H^{s})$, and the solution depends continuously on the initial value. Meanwhile, the maximum existence time of the solution and its growth restriction are given.For global well - posedness, when $s\geq2$, through energy estimates and local theory, combined with conservation laws, it is proved that the initial - value problem of the equation is globally well - posed in $H^{s}(\mathbb{R})$. When $1\leq s<2$, the initial value is decomposed into a rough small part and a smooth part, and evolution equations are established respectively. It is proved that the corresponding integral equation is locally well - posed in $H^{2}$ and the solution can be extended, thus concluding that the initial - value problem of the equation is globally well - posed in $H^{s}$.