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Abstract
The classical Benjamin–Bona–Mahony equation (BBM equation) models unidirectional propagation of long gravity surface waves of small amplitude. Unlike many other water wave models, it lacks the Galilean invariance, which is an essential property of physical systems. It is shown that by an addition of a higher asymptotic order nonlinear term, this deficiency can be corrected, giving rise to a new Galilei invariant Benjamin–Bona–Mahony equation (iBBM equation). Moreover, further additional higherorder terms can be chosen in a way that the augmented model preserves the energy conservation property along with Hamiltonian and Lagrangian structures. The resulting equation is referred to as energypreserving Benjamin–Bona–Mahony equation (eBBM). It is shown that both the classical BBM equation and the energypreserving eBBM equations belong to a oneparameter (α) family that shares essentially the same local and nonlocal symmetries, conservation laws, Hamiltonian, and Lagrangian structures, with the BBM and eBBM equations corresponding to parameter values α=0 and α=1, respectively. Symmetry and conservation law classifications reveal a special case α=1/3, which is shown to correspond to a rescaled version of the celebrated integrable Camassa–Holm (CH) equation. Local symmetries and conservation laws are computed, and numerical solution behaviour is compared for the three BBMtype modes and the CHequivalent eBBM_{1/3} model.
Original language  British English 

Article number  100519 
Journal  Partial Differential Equations in Applied Mathematics 
Volume  7 
DOIs  
State  Published  Jun 2023 
Keywords
 Benjamin–Bona–Mahony equation
 Conservation laws
 Energy conservation
 Galilean invariance
 Nonlinear dispersive waves
 Symmetry
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FSU2023014 [D. Dutykh]  Nonlinear waves in geophysics and biomechanics
Dutykh, D. (PI)
1/02/23 → 31/01/25
Project: Research