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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 higher-order 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 energy-preserving Benjamin–Bona–Mahony equation (eBBM). It is shown that both the classical BBM equation and the energy-preserving eBBM equations belong to a one-parameter (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 BBM-type modes and the CH-equivalent eBBM1/3 model.
Original language | British English |
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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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Dive into the research topics of 'Galilei-invariant and energy-preserving extensions of Benjamin–Bona–Mahony-type equations'. Together they form a unique fingerprint.Projects
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FSU-2023-014 [D. Dutykh] - Nonlinear waves in geophysics and biomechanics
Dutykh, D. (PI)
1/02/23 → 31/01/25
Project: Research