Benjamin–Bona–Mahony equation

The Benjamin–Bona–Mahony equation (or BBM equation) – also known as the regularized long-wave equation (RLWE) – is the partial differential equation

This equation was studied in Benjamin, Bona, and Mahony (1972) as an improvement of the Korteweg–de Vries equation (KdV equation) for modeling long surface gravity waves of small amplitude – propagating uni-directionally in 1+1 dimensions. They show the stability and uniqueness of solutions to the BBM equation. This contrasts with the KdV equation, which is unstable in its high wavenumber components. Further, while the KdV equation has an infinite number of integrals of motion, the BBM equation only has three.

Before, in 1966, this equation was introduced by Peregrine, in the study of undular bores.

A generalized n-dimensional version is given by

where ${\displaystyle \varphi }$ is a sufficiently smooth function from ${\displaystyle \mathbb {R} }$ to ${\displaystyle \mathbb {R} ^{n}}$. Avrin & Goldstein (1985) proved global existence of a solution in all dimensions.

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