Multiple solutions and conserved vectors of a shallow water wave equation arising in fluid mechanics; Lie group analysis

Abstract

Investigation has shown that shallow water wave model equations constitute a class of hyperbolic partial differential equations describing the movement below a pressure surface in a fluid, and this makes them highly useful to be studied. Therefore, in this paper we analytically investigate a one-dimensional shallow water wave equation that is highly applicable in fluid dynamics. We leverage the robust nature of the Lie group technique to provide symmetries to the equation. Commutator and adjoint representations of symmetries were generated and presented in tabular form. Sequel to that, we perform reductions of the shallow water wave model under study via its point symmetries so that its exact solutions can be found. Thus, to achieve this, various combinations of the symmetries are made. Besides, we examine the travelling wave solutions of the model. The direct integration method is used to obtain elliptic integral functions as well as elliptic solutions in terms of the Weierstrass function. In addition, we utilize standard techniques such as (G′/G)-expansion, power series, and Kudryashov’s approaches to generate more closed-form solutions to the model. Other interesting solutions achieved include hyperbolic, trigonometric, rational, and exponential functions. In search of the physical meaning associated with the results, graphical demonstrations are included, and these are given in 2D, 3D, and density plots. Thereafter, the construction of conserved vectors for the aforementioned model is done with the use of the standard multiplier approach and Ibragimov’s theorem. These techniques give the opportunity to furnish various forms of conserved currents.