Lie group analysis for obtaining the abundant group invariant solutions and dynamics of solitons for the Lonngren-wave equation

The Lonngren-wave equation (LW Equation), one of the many nonlinear evolution equations (N-EEs) that arise in the field of mathematical physics, is the subject of this study, which uses an extremely strong analytical technique known as "Lie group analysis"to create novel solutions. We obtain a five-dimensional optimal system based on four-dimensional Lie algebra. We compute the group invariant solutions via subalgebras. Our obtained solutions are based on the trigonometric, hyperbolic, and polynomial functions. By varying the parameters, solutions exhibit wavelike properties that include bright, dark, singular, dark-singular-combined solitons, periodic singular and dark-bright-combined. The physical dynamics of the obtained solutions are explored by the 3D and 2D Mathematica simulations which are explaining new properties of the model considered in this paper.