Abstract
The Benjamin–Bona–Mahony equation describes the unidirectional propagation of small-amplitude long waves on the surface of water in a channel. In this paper, we consider a family of generalized Benjamin–Bona–Mahony–Burgers equations depending on three arbitrary constants and an arbitrary function G(u). We study this family from the standpoint of the theory of symmetry reductions of partial differential equations. Firstly, we obtain the Lie point symmetries admitted by the considered family. Moreover, taking into account the admitted point symmetries, we perform symmetry reductions. In particular, for G′(u)≠0, we construct an optimal system of one-dimensional subalgebras for each maximal Lie algebra and deduce the corresponding (1+1)-dimensional nonlinear third-order partial differential equations. Then, we apply Kudryashov’s method to look for exact solutions of the nonlinear differential equation. We also determine line soliton solutions of the family of equations in a particular case. Lastly, through the multipliers method, we have constructed low-order conservation laws admitted by the family of equations.
Highlights
The Benjamin–Bona–Mahony equation (BBM), or regularised long wave (RLW) equation, Academic Editors: Michael ut + u x + uu x − u xxt = 0, Tsamparlis and Andronikos (1) PaliathanasisReceived: 2 August 2021Accepted: 26 October 2021Published: 3 November 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in was first proposed by Benjamin et al [1] as an alternative mathematical model to the Korteweg–de Vries for modelling long wave motions in nonlinear dispersive systems.The authors stressed that both models are applicable at the same level of approximation, from a computational mathematics point of view, the BBM equation has some advantages over the KdV equation
We consider a generalized family of Benjamin–Bona–Mahony–Burgers equations given by published maps and institutional affiliations
The method proposed in [39] gives an explicit algorithm for partial differential equation (PDE) with n ≥ 2 independent variables admitting a symmetry algebra whose dimension is at least n − 1 that allows us to find all symmetry-invariant conservation laws, which will reduce to first integrals for the ordinary differential equations (ODEs) that describe the symmetry-invariant solutions of the PDE
Summary
The analysis of Lie symmetries is one of the most effective algorithms to analyse PDE equations including the construction of invariant solutions, construction of mappings between equivalent equations of the same family, finding invertible mappings of nonlinear PDEs to linear PDEs or finding conservation laws. The method proposed in [39] gives an explicit algorithm for PDEs with n ≥ 2 independent variables admitting a symmetry algebra whose dimension is at least n − 1 that allows us to find all symmetry-invariant conservation laws, which will reduce to first integrals for the ODE that describe the symmetry-invariant solutions of the PDE.
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