Advances in Mathematical Physics

Volume 2018, Article ID 4743567, 11 pages

https://doi.org/10.1155/2018/4743567

## Approximate Symmetry Analysis and Approximate Conservation Laws of Perturbed KdV Equation

Department of Mathematics, Inner Mongolia University of Technology, Hohhot, 010051, China

Correspondence should be addressed to Yu-Shan Bai; nc.ude.tumi@nahsuyiabm

Received 9 February 2018; Revised 11 August 2018; Accepted 27 August 2018; Published 9 September 2018

Academic Editor: Boris G. Konopelchenko

Copyright © 2018 Yu-Shan Bai and Qi Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Approximate symmetries, which are admitted by the perturbed KdV equation, are obtained. The optimal system of one-dimensional subalgebra of symmetry algebra is obtained. The approximate invariants of the presented approximate symmetries and some new approximately invariant solutions to the equation are constructed. Moreover, the conservation laws have been constructed by using partial Lagrangian method.

#### 1. Introduction

The partial differential equations (PDEs) with small parameter have been arisen in mathematics, physics, mechanics, etc. For these perturbed equations, the finding of analytic solutions, conservation laws, symmetries, etc. is essential in these related fields. Traditionally, various perturbation methods are used to solve such equations. In addition, emerging innovative methods such as the inverse scattering transformation method, homotopy perturbation method, Adomian decomposition method, and approximate symmetry method have been developed fast in the past few decades [1–4]. The approximate symmetry method, which is developed by Baikov, Gazizov, and Ibragimov [3, 4] in the 1980s, shows an effectiveness to obtain approximate solutions to a perturbed PDEs. The idea behind this development was the extension of Lie’s theory in which a small perturbation of the original equation is encountered. The new method maintains the essential features of the standard Lie symmetry method and provides us with the most widely applicable technique to find the approximate solutions to a perturbed differential equations [5, 6].

The construction of conservation laws is one of the key applications of symmetries to physical problems [7, 8]. But for perturbed systems, one hindrance is how to construct conservation laws. Combined with the Noether theory and Lagrangian principle, the approximate symmetry method yields the approximate conservation laws for a perturbed PDEs. There are mainly two methods. One approach, based on generalizations of the Noether’s theorem to perturbed equations, is given to get approximate conservation laws via the approximate Noether symmetries associated with a Lagrangian of the perturbed equations [5, 9]. This method depends on the existence of the Lagrangian functional for underlying differential equations. The other approach (also called partial Lagrangian method), which is presented by Johnpillai, Kara, and Mahomed [10, 11], demonstrates how one can construct approximate conservation laws of Eular-Type PDEs via approximate Noether-type symmetry operators associated with partial Lagrangian. Partial Lagrangian method can construct approximate conservation laws of perturbed equations via approximate operators that are not necessarily approximate symmetry operators of the underlying system of equations. This method is used for more general equations as even the equations do not admit essential Lagrangian.

In this paper, we consider perturbed KdV equationwhere is a small parameter, while and are arbitrary constants. Obviously, the equation has no Lagrangian because of its odd order. In [12], E. S. Benilov and B. A. Malomed discussed the equation based on the inverse scattering transformation and showed its integrability. When (1) has important applications in the description of nonlinear ion acoustic waves in an inhomogeneous plasma. For many other applications of (1), please refer to [2, 12, 13] and the references therein.

In this article, with the application of approximate symmetry method and partial Lagrangian method, we will investigate (1) and show its all first-order approximate symmetries. Furthermore, we will also construct several approximate solutions and approximate conservation laws of the equation. The rest of the paper is organized as follows. Section 2 gives some basic concepts and notations. Section 3 analyzes approximate symmetries of (1) by applying the approximate symmetry method, developed by Baikov, Gazizov, and Ibragimov. We compute the optimal system of the presented approximate symmetries. In Section 4 we generate the approximate invariants of presented approximate symmetries and construct corresponding approximately invariant solutions. Section 5, the final part, presents approximate conservation laws via partial Lagrangian method.

#### 2. Notations and Definitions

We will use the following notations and definitions. Let be a one-parameter approximate transformation group:An approximate equationis said to be* approximately invariant* with respect to or admits ifwhenever satisfies (3).

If , where independent variables , dependent variables and denote the collections of all th-order partial derivatives, then (3) becomes an approximate differential equation of , and is an approximate symmetry group of the differential equation.

Theorem 1 (see [5]). *Equation (3) is approximately invariant under the approximate transformation group (2) with the generatorif and only iforin which is order of equation and is order prolongation of . The operator (5) satisfying (7) is called an infinitesimal approximate symmetry or an approximate operator admitted by (3). Accordingly, (7) is termed the determining equation for approximate symmetries.*

*Remark 2. *The determining equation (7) can be written as follows:The factor is determined by (8) and then substituted in (9). The latter equation must hold for all solutions of . Comparing (8) with the determining equation of exact symmetries, we obtain the following statement.

Theorem 3 (see [5]). *If (3) admits an approximate transformation group with the generator , where , then the operatoris an exact symmetry of *

*Remark 4. *It is manifested from (8) and (9) that if is an exact symmetry of (11), then is an approximate symmetry of (3).

*Definition 5 (see [5]). *Equations (11) and (3) are termed an unperturbed equation and a perturbed equation, respectively. Under the conditions of Theorem 3, the operator is called a stable symmetry of the unperturbed equation (11). The corresponding approximate symmetry generator for the perturbed equation (10) is called a deformation of the infinitesimal symmetry of (11) caused by the perturbation . In particular, if the most general symmetry Lie algebra of Eq.(11) is stable, we say that the perturbed equation (3) inherits the symmetries of the unperturbed equation.

#### 3. Approximate Symmetry Analysis

##### 3.1. Exact Symmetries

Let us write the approximate group generator in the formwhere , , and are unknown functions of , , and .

Solving the determining equationfor the exact symmetries of the unperturbed equation , we can getThen, we obtain , where are arbitrary constants. Hence,In other words, admits the four-dimensional Lie algebra with the basis

##### 3.2. Approximate Symmetries and Optimal System

###### 3.2.1. Approximate Symmetries: The Cases and Are Arbitrary

First we need to determine the auxiliary function by virtue of (15), by Substituting the expression (15) of the generator into above equation we obtain the auxiliary function:Now, calculate the operators by solving the inhomogeneous determining equation for deformations: Above determining equation yieldsSolving this system, we obtain Then, we obtain the following approximate symmetries of (1)and we have

Tables 1, 2, and 3 of commutator, evaluated in the first-order of precision, show that above operators span a seven-dimensional approximate Lie algebra