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

,e Korteweg–de Vries (KdV) equation is a weakly nonlinear third-order differential equation which models and governs the evolution of fixed wave structures. ,is 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. ,e 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.


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 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. e 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. e 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]. e 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.
is 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. e method to handle the approximate part of the KdV equation is developed in Section 3. e 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. e work is concluded by describing the highlights in Section 5.

Exact Symmetries and Conservation Laws of the Korteweg-de Vries (KdV) Equation
e exact symmetries and conservation laws in the current study for the work considered in [8,9] are worked out as follows: e Korteweg-de Vries (KdV) equation which is a thirdorder nonlinear partial differential equation is (1) e infinitesimal symmetry operator is (2) Applying this symmetry operator on (1), we get e expanded form of equation (4) is Substituting equation (1) in equation (5), we get 2

Journal of Mathematics
Comparing the coefficients of various terms, we get the coefficients and monomials, as shown in Table 1. Table 1 yields the required set of PDEs as follows: Form (10), As therefore, Integrating twice with respect to t yields From (10), Integrating with respect to "x," From (13), Hence, the Lie symmetry generators for the KdV equation are given, as shown in Table 2.

A New Procedure to Find the Approximate Symmetries
is section explains the development of the method for the approximate symmetries of the KdV equation. e KdV (1) where ε is a small parameter, causing the required perturbation in the KdV equation. e exact and approximate parts of (22) are Journal of Mathematics 3 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, and is the approximate Lie symmetry generator. Furthermore, ϕ, ϱ, and φ are the unknown functions of x, t, and μ, respectively. Now, applying the generator X on (24), we have which yields e comparison of coefficients of ε 0 and ε 1 , respectively, yields the exact and approximate symmetries of the corresponding PDEs as in the following: (30) e 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

Approximate Symmetries and Corresponding Conservation Laws of the KdV Equation
In this section, we apply the developed method to find out the approximate symmetries. is method is applied and discussed for different cases. Considering the perturbed KdV equation [6,11,12], By employing the method developed in [13][14][15] for the expansion of μ, Using this expansion in (31), Table 1: e exact symmetries of the given partial differential equation (PDE).

Coefficients
Monomials Lie symmetry generators Journal of Mathematics m t + εn t − 6(m + εn) m x + εn x + m xxx + εn xxx � εf x, y, n, t, n t , n x , m t , m x , Equation (33) in more compact form is (neglecting higher power of ϵ ) e comparison of the coefficients of ϵ 0 and ϵ 1 in (33) gives e Lie symmetry generator is Here, Applying the Lie generator, X Δ e + εΔ a � 0, which gives us We now discuss the following cases in a bit detail.
Case I. Let f x, y, n, t, n t , n x , m t , m x � − m t − n t .
en, determining the system of PDEs from (35), As zϱ zm � 0, Journal of Mathematics 5 which implies that "ϱ" is the function of "t" alone. erefore, Integrating the above equation twice with respect to "t" yields Also, which shows that "ϕ" is the function of "x" alone. erefore, Putting the value of "ϱ t " in (46), we get Integrating (47), we get Now, Putting the value of "ϱ t " in (49), By taking and putting the value of "ϱ t " in (51), e corresponding symmetry generators are tabulated in Table 3.

Conservation Laws.
e conservation laws are developed as in the following: X 1 ψ x, y, n, t, n t , n x , m t , m x � 0, Now, by taking Furthermore, X 2 ψ x, y, n, t, n t , n x , m t , m x � 0, (57) Following are the symmetries and their corresponding conservation laws of Case 1. (59) Applying (36) to (59) yields the following system of PDEs: (60) Solving the above system of PDEs, we get the following results: e approximate symmetries and their corresponding conservation laws in this case are given in Table 4.
Case 3. For this case, take f x, y, n, t, n t , n x , m t , m x � − n x . (62) From (35), we get after comparing the coefficients of ϵ 0 and ϵ 1 , is results in the following equations: Following are the symmetries and corresponding conservation laws of this Case 3.
then the system defined in (35) gives Applying (36) to (66), we get the following set of PDEs: Lie symmetry generators e above equations yield Following are the symmetries and corresponding conservation laws of Case 4.

Case 5.
Let f x, y, n, t, n t , n x , m t , m x � − n, then the system defined in (35) gives Applying (36) to (70) produces the following set of PDEs: Solving the above equations, we get Following are the symmetries and corresponding conservation laws of Case 5.
Lie symmetry generators Corresponding conservation laws ψ 2 � f(x, y, n, n t , n x , m t , m x ) X 3 � (z/zx) ψ 3 � g(y, n, t, n t , n x , m t , m x ) Table 7: Lie symmetry generators and corresponding conservation laws.

Conclusion
e 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. e Lie symmetries along with their conservation laws are given in Tables 2, 4  is 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 2-6. 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.

Data Availability
ere are no specific data used in the study of this article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.