Eur. Phys. J. Plus (2022) 137: 401
https://doi.org/10.1140/epjp/s13360-022-02619-5

Regular Article

Some new wave profiles and conservation laws in a Pre-compressed one-dimensional granular crystal by Lie group analysis

¹ Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowskiego St., 90-924, Lodz, Poland
² Department of Mathematics, University of Management and Technology, Lahore, Pakistan
³ Department of Mathematics, Namal University, 30 Km Talagang Road, 42250, Mianwali, Pakistan
⁴ Department of Mathematics and Statistics, College of Science, Taif University, P. O. Box 11099, 21944, Taif, Saudi Arabia

Received: 18 August 2021
Accepted: 16 March 2022
Published online: 29 March 2022

Abstract

In the present exploration, some new wave profiles are constructed via Lie symmetry of transformations scheme by the one-dimensional optimal system in a pre-compressed one-dimensional granular crystal. Lie group scheme is used to construct the infinitesimal generators, entire vector field, commutation relations, and adjoint representations. Infinitesimal generators are used to construct one optimal system of the one-dimensional sub-algebras. Employing the optimal system, similarity reductions are obtained for the considered nonlinear model. Translational vectors and their linear combination are used to covert the assumed model in a nonlinear ordinary differential equation. These methods give us some new and interesting solutions which contain the trigonometric, hyperbolic trigonometric, and exponential type functions. To represent the physical significance of the assumed model, some 3D and 2D diagrams of acquired results are plotted by using Mathematica under the suitable choice of involving parameters values. The multiplier scheme is employed to describe the conserved vectors of the assumed problem.