On Galilean Invariant and Energy Preserving BBM-Type Equations

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.

Keywords

• nonlinear dispersive waves
• Benjamin–Bona–Mahony equation%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <span style="font-variant: small-caps">Benjamin–Bona–Mahony</span> equation
• conservation laws
• symmetries