Dynamics of analytical solutions to Benny–Luke dispersive wave occurring at a beach

The aim of this research paper is to derive a variety of some novel analytical solutions to the Benny–Luke Equation by using the similarity transformations method via Lie-symmetry analysis. The equation is a nonlinear dispersive wave occurring at a beach. It results after the interaction of long waves of small amplitudes in a finite-depth water wave, and exists where surface tension is near to zero. The solutions are derived with an appropriate choice of arbitrary constants to proceed integration in the similarity reduction. A numerical simulation of the solutions is also performed in order to show their dynamics. The dynamical behavior of solutions reveals traveling, stationary, periodic, and parabolic profiles based on their graphical representations. Novelty of solutions confirmed by comparing them with the results established in reported works.