Exact Solutions and Conservation Laws of a Generalized (1 + 1) Dimensional System of Equations via Symbolic Computation

Sivenathi Oscar Mbusi,Ben Muatjetjeja,Abdullahi Rashid Adem

doi:10.3390/math9222916

Sivenathi Oscar Mbusi, Ben Muatjetjeja + Show 1 more

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https://doi.org/10.3390/math9222916

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Abstract

The aim of this paper is to compute the exact solutions and conservation of a generalized (1 + 1) dimensional system. This can be achieved by employing symbolic manipulation software such as Maple, Mathematica, or MATLAB. In theoretical physics and in many scientific applications, the mentioned system naturally arises. Time, space, and scaling transformation symmetries lead to novel similarity reductions and new exact solutions. The solutions obtained include solitary waves and cnoidal and snoidal waves. The familiarity of closed-form solutions of nonlinear ordinary and partial differential equations enables numerical solvers and supports stability analysis. Although many efforts have been dedicated to solving nonlinear evolution equations, there is no unified method. To the best of our knowledge, this is the first time that Lie point symmetry analysis in conjunction with an ansatz method has been applied on this underlying equation. It should also be noted that the methods applied in this paper give a unique solution set that differs from the newly reported solutions. In addition, we derive the conservation laws of the underlying system. It is also worth mentioning that this is the first time that the conservation laws for the equation under study are derived.

Highlights

The illustrious equation ut + 6uu x + u xxx = 0Received: 26 August 2021Accepted: 1 November 2021Published: 16 November 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. (1)is an example of a nonlinear evolution equation (NLEE) [1,2]
The nonlinear term uu x accounts for the steepening of the wave [5,6], whereas the linear dispersive term u xxx describes the spreading of the wave [7,8]. This essential equation illustrates the subtleties of solitary waves
Equation (1) is a significant equation in the theory of integrable systems since it has an infinite number of conservation laws, multiple-soliton solutions, and many other physical properties [1]

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. The term ut describes the time evolution of the wave [3,4] and, as such, (1) is considered an evolution equation. The nonlinear term uu x accounts for the steepening of the wave [5,6], whereas the linear dispersive term u xxx describes the spreading of the wave [7,8] This essential equation illustrates the subtleties of solitary waves. The mKdV equation yields algebraic soliton solutions in the form of rational functions. To the best of our knowledge, this is the first time that Lie point symmetry analysis in conjunction with an ansatz method has been applied to this underlying equation. The infinitesimal point symmetries of (4) form the three-dimensional Lie algebra spanned by the following linearly independent operators: Θ1.

Exact Solutions Using an Anstaz Method

B2 2 α

B2 2 β

Conservation Laws

Concluding Remarks