Approximate Symmetries Analysis and Conservation Laws Corresponding to Perturbed Korteweg–de Vries Equation

Tahir Ayaz,Zabidin Salleh,Wali Khan Mashwani,Ikramullah Ikramullah,Israr Ali Khan,Farhad Ali,Zakia Hammouch

doi:10.1155/2021/7710333

Abstract

The Korteweg–de Vries (KdV) equation is a weakly nonlinear third-order differential equation which models and governs the evolution of fixed wave structures. This paper presents the analysis of the approximate symmetries along with conservation laws corresponding to the perturbed KdV equation for different classes of the perturbed function. Partial Lagrange method is used to obtain the approximate symmetries and their corresponding conservation laws of the KdV equation. The purpose of this study is to find particular perturbation (function) for which the number of approximate symmetries of perturbed KdV equation is greater than the number of symmetries of KdV equation so that explore something hidden in the system.

Highlights

Differential equations (DEs) are ubiquitous in modeling an extensive class of physical phenomena involving variation with respect to one or more independent variables. erefore, DEs are broadly divided into ordinary DEs (ODEs) and partial DEs (PDEs)
Modeling of PDEs under special conditions and constraints is advantageous in different situations for an effective manipulation of the varying phenomenon. e majority of real-world problems are almost nonlinear in nature, having no analytical solutions
Conclusion e Korteweg–de Vries (KdV) equation is a 3rd order nonlinear partial differential equation which is modeled for waves on the surface of shallow water

Summary

Introduction

Differential equations (DEs) are ubiquitous in modeling an extensive class of physical phenomena involving variation with respect to one or more independent variables. erefore, DEs are broadly divided into ordinary DEs (ODEs) and partial DEs (PDEs). In order to solve nonlinear problems, various approximations and techniques are used to gain high accuracy In this regard, the approximate symmetry methods play a significant role. A perturbed nonlinear wave equation is a class of approximate symmetries which is computed using two newly developed methods. Substituting equation (1) in equation (5), we get φt − φtμx +􏼐φμ − ρt􏼑μt − φμμxμt − ρμμ2t − 6φμx − 6μ􏽨φx +􏼐φμ − φx􏼑μx − ρxμt − φμμ2x − ρμμxμt􏽩 + φxxx +􏼐3φxxμ − φxxx􏼑μx − ρxxxμt +􏼐3φxμμ − 3φxxμ􏼑μ2x − 3ρxxμμxμt +􏼐3φxμ − 3φxx􏼑μxx − 3ρxxμxt +􏼐φμμμ − φxμμ􏼑μ3x − 3ρxμμμ2xμt (6) +􏼐3φμμ − 9φxμ􏼑μxμxx − 6ρxμμxμxt − 3ρxμμtμxx + 6μ􏼐φμ − 3φx􏼑μx − 􏼐φμ − 3φx􏼑μt − 3ρxμxxt − φμμμμ4x − ρμμμμ3xμt − 6φμμμ2xμxx − 3ρμμμ2xμxt − 3ρμμμxμtμxx − 24μφμμ2x + 4φμμxμt − 3ρμμxμxxt − 3φμμ2xx − 3ρμμxxμxt − 6μρμμxμxt + ρμμ2t 0.

Integrating twice with respect to t yields

Lie symmetry generators

Xe φezx