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Volume 2021 |Article ID 7710333 | https://doi.org/10.1155/2021/7710333

Tahir Ayaz, Farhad Ali, Wali Khan Mashwani, Israr Ali Khan, Zabidin Salleh, Ikramullah, "Approximate Symmetries Analysis and Conservation Laws Corresponding to Perturbed Korteweg–de Vries Equation", Journal of Mathematics, vol. 2021, Article ID 7710333, 11 pages, 2021. https://doi.org/10.1155/2021/7710333

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

Academic Editor: Zakia Hammouch
Received25 Apr 2021
Accepted10 Jun 2021
Published22 Jun 2021

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.

1. Introduction

Differential equations (DEs) are ubiquitous in modeling an extensive class of physical phenomena involving variation with respect to one or more independent variables. Therefore, DEs are broadly divided into ordinary DEs (ODEs) and partial DEs (PDEs). In different sectors of science and technology, PDEs have played a significant role. PDEs have numerous applications in mathematics, physics, fluid dynamics, mechanics, and physical chemistry. Modeling of PDEs under special conditions and constraints is advantageous in different situations for an effective manipulation of the varying phenomenon. The majority of real-world problems are almost nonlinear in nature, having no analytical solutions. 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. We have used the method of approximate Lie symmetry [1, 2], for PDEs to deal with the dynamical system more accurately. In the 1980s, the method of approximate Lie symmetry was developed by Baikov et al. [3, 4]. In obtaining the approximate solutions to such perturbed PDEs, the approximate symmetry method is an effective one. The extension of Lie’s theory was mainly the basic reason behind the development of approximate symmetry, which deals with the systems by introducing small perturbation [5]. Symmetry applications to physical problems play a pivotal role in the development of conservation laws [6, 7]. The widely recognized KdV equation is a mathematical model for the depiction of weak nonlinear long wavelength waves in various branches of engineering and physics. It explains how waves evolve due to comparable effects of weak nonlinearity and dispersion. A perturbed nonlinear wave equation is a class of approximate symmetries which is computed using two newly developed methods. For both methods, the associated invariant solution with the approximate symmetries is constructed. By discussing the advantages and disadvantages of each method, the symmetries and solutions are compared. So, the Lie group technique in finding the exact solution of a differential equation has lost its importance. But an approximate Lie group technique has been implemented and used in various methods for obtaining additional related information of differential equation. Perturbation analysis is one of the techniques which is used particularly for nonlinear systems.

This study is framed in the following manner: Section 2 is devoted to the development of exact symmetries and exact conservation laws of the KdV equation. The method to handle the approximate part of the KdV equation is developed in Section 3. The method so developed is applied to tackle the approximate part of the KdV equation for different cases and their corresponding conservation laws in Section 4. The work is concluded by describing the highlights in Section 5.

2. Exact Symmetries and Conservation Laws of the Korteweg–de Vries (KdV) Equation

The exact symmetries and conservation laws in the current study for the work considered in [8, 9] are worked out as follows:

The Korteweg–de Vries (KdV) equation which is a third-order nonlinear partial differential equation is

The infinitesimal symmetry operator is

Applying this symmetry operator on (1),we get

The expanded form of equation (4) is

Substituting equation (1) in equation (5), we get

Comparing the coefficients of various terms, we get the coefficients and monomials, as shown in Table 1.


CoefficientsMonomials

1

Table 1 yields the required set of PDEs as follows:

Form (10),

Astherefore,

Let

Integrating twice with respect to yields

From (10),

Integrating with respect to “,

From (13),

The general solution is

Hence, the Lie symmetry generators for the KdV equation are given, as shown in Table 2.


Lie symmetry generators


3. A New Procedure to Find the Approximate Symmetries

This section explains the development of the method for the approximate symmetries of the KdV equation. The KdV (1) is perturbed with the function aswhere is a small parameter, causing the required perturbation in the KdV equation. The exact and approximate parts of (22) are

Equation (22) can now be written in a more compact form as

On similar footing, we can combine the exact and approximate Lie symmetries as

Here,is the exact Lie symmetry generator, andis the approximate Lie symmetry generator. Furthermore, , , and are the unknown functions of , and , respectively.

Now, applying the generator on (24), we havewhich yields

The comparison of coefficients of and , respectively, yields the exact and approximate symmetries of the corresponding PDEs as in the following:

The latter equation additionally gives the approximate Lie symmetries, which will not only provide the approximate conservation laws involved in the dynamics of the KdV equation but will also give the unknown function [8, 10].

4. Approximate Symmetries and Corresponding Conservation Laws of the KdV Equation

In this section, we apply the developed method to find out the approximate symmetries. This method is applied and discussed for different cases. Considering the perturbed KdV equation [6, 11, 12],

By employing the method developed in [1315] for the expansion of ,

Using this expansion in (31),

Equation (33) in more compact form is (neglecting higher power of )

The comparison of the coefficients of and in (33) gives

The Lie symmetry generator is

Here,

Applying the Lie generator,which gives us

We now discuss the following cases in a bit detail.

Case I. LetThen, determining the system of PDEs from (35),Aswhich implies that “” is the function of “” alone. Therefore,Integrating the above equation twice with respect to “” yieldsAlso,which shows that “” is the function of “” alone. Therefore,Putting the value of “” in (46), we getIntegrating (47), we getNow,Putting the value of “” in (49),By takingand putting the value of “” in (51),Therefore,The corresponding symmetry generators are tabulated in Table 3.


Lie symmetry generators


4.1. Conservation Laws

The conservation laws are developed as in the following:

Now, by takingso

Furthermore,

Following are the symmetries and their corresponding conservation laws of Case 1.

Case 2. LetFrom (35), we get after comparing the coefficients of and ,Applying (36) to (59) yields the following system of PDEs:Solving the above system of PDEs, we get the following results:The approximate symmetries and their corresponding conservation laws in this case are given in Table 4.


Lie symmetry generatorsCorresponding conservation laws


Case 3. For this case, takeFrom (35), we get after comparing the coefficients of and ,This results in the following equations:Following are the symmetries and corresponding conservation laws of this Case 3.

Case 4. For this case, takethen the system defined in (35) givesApplying (36) to (66), we get the following set of PDEs:The above equations yieldFollowing are the symmetries and corresponding conservation laws of Case 4.

Case 5. Letthen the system defined in (35) givesApplying (36) to (70) produces the following set of PDEs:Solving the above equations, we getFollowing are the symmetries and corresponding conservation laws of Case 5.

Case 6. Assumethen the system defined in (35) givesApplying (36) to (74) results in the following set of PDEs:Solving the above set of equations, we getFollowing are the symmetries and corresponding conservation laws of Case 6.

5. Conclusion

The KdV equation is a 3rd order nonlinear partial differential equation which is modeled for waves on the surface of shallow water. It admits four Lie symmetries given in Table 2. In this paper, approximate symmetry techniques are used for finding some classes of the KdV equations that admit more symmetries as compared to the exact KdV equations. We perturbed the KdV equation by different particular functions and found the corresponding Lie symmetries. We found two important classes for the perturbed KdV equation that admits five Lie symmetries. The Lie symmetries along with their conservation laws are given in Tables 2, 4 and 5. In both the tables, we have an extra symmetry which corresponds to an extra conservation law. This extra conservation law is an extra information hidden in the system, the perturbation procedure explored it. Sometimes, the symmetry does not exist for the exact equation, but perturbation enables the equation to admit a symmetry. We saw this phenomenon in this research work by comparing Tables 26. Table 1 contains the determining PDEs which provide the set of Lie symmetries admitted by the given PDE. We have 4 Lie symmetries given in Table 2 for exact PDE, while Tables 3 and 6 contain only three Lie symmetries; in these cases, we lose one symmetry (one conservation law). Tables 7 and 8 consist of four Lie symmetries which means that all the conservation laws are recovered in these cases. Table 9 includes the lie symmetry generators and corresponding conservation laws.


Lie symmetry generatorsCorresponding conservation laws



Lie symmetry generatorsCorresponding conservation laws



Lie symmetry generatorsCorresponding conservation laws