Abstract
We investigate a family of higher-order Benjamin–Bona–Mahony-type equations, which appeared in the course of study towards finding a Galilei-invariant, energy-preserving long wave equation. We perform local symmetry and conservation laws classification for this family of Partial Differential Equations (PDEs). The analysis reveals that this family includes a special equation which admits additional, higher-order local symmetries and conservation laws. We compute its solitary waves and simulate their collisions. The numerical simulations show that their collision is elastic, which is an indication of its S−integrability. This particular PDE turns out to be a rescaled version of the celebrated Camassa–Holm equation, which confirms its integrability.
Highlights
The famous B ENJAMIN –B ONA –M AHONY (BBM) equation was derived for the first time D
The first reason for that is that the potential α-family Equation (8) are nonlocally related to the Partial Differential Equations (PDEs) (2), and the local symmetries and conservation laws of (2) vs
A family of PDEs (2) with one free parameter was considered. This family was inspired by our quest for G ALILEI-invariant and energy-preserving higher order analogs of the classical BBM equation [2]
Summary
The famous B ENJAMIN –B ONA –M AHONY (BBM) equation was derived for the first time D. We study symmetries, conservation laws, and solitary wave solutions to family (2). The conservation laws of a variable coefficients BBM equation were studied in [6]. When α = 0 , the generator X 3 yields the scaling symmetry group holding for the BBM Equation (1), and when α = 1 , X 3 yields the G ALILEI group. The latter case is the only G ALILEI-invariant representative of the PDE family (2).
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