The Benjamin–Bona–Mahony equation (or BBM equation) is the partial differential equation

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

This equation was introduced in as an improvement of the Korteweg–de Vries equation

Korteweg–de Vries equation

In mathematics, the Korteweg–de Vries equation is a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an exactly solvable model, that is, a non-linear partial differential equation whose solutions can be exactly and precisely specified...

 (KdV equation) for modeling long waves of small amplitude 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

Wavenumber

In the physical sciences, the wavenumber is a property of a wave, its spatial frequency, that is proportional to the reciprocal of the wavelength. It is also the magnitude of the wave vector...

 components. Further, while the KdV equation has an infinite number of integrals of motion, the BBM equation only has three.

A higher-dimensional version is given by


where is a fixed smooth function from to . proved local existence of a solution in all dimensions.