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 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 φ {displaystyle varphi } is a sufficiently smooth function from R {displaystyle mathbb {R} } to R n {displaystyle mathbb {R} ^{n}} . Avrin & Goldstein (1985) proved global existence of a solution in all dimensions. The BBM equation possesses solitary wave solutions of the form: where sech is the hyperbolic secant function and δ {displaystyle delta } is a phase shift (by an initial horizontal displacement). For | c | < 1 {displaystyle |c|<1} , the solitary waves have a positive crest elevation and travel in the positive x {displaystyle x} -direction with velocity 1 / ( 1 − c 2 ) . {displaystyle 1/(1-c^{2}).} These solitary waves are not solitons, i.e. after interaction with other solitary waves, an oscillatory tail is generated and the solitary waves have changed. The BBM equation has a Hamiltonian structure, as it can be written as: Here δ H / δ u {displaystyle delta H/delta u} is the variation of the Hamiltonian H ( u ) {displaystyle H(u)} with respect to u ( x ) , {displaystyle u(x),} and ∂ x {displaystyle partial _{x}} denotes the partial differential operator with respect to x . {displaystyle x.}