Advances in Mathematical Physics
Volume 2018 (2018), Article ID 4743567, 11 pages
https://doi.org/10.1155/2018/4743567

Research Article

Approximate Symmetry Analysis and Approximate Conservation Laws of Perturbed KdV Equation

Yu-Shan Bai and Qi Zhang

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

Correspondence should be addressed to Yu-Shan Bai; mbaiyushan@imut.edu.cn

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 equation

where 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 equation

is said to be approximately invariant with respect to or admits if

whenever 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 generator

if and only if

or

in 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 operator

is 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 form

where , , and are unknown functions of , , and .

Solving the determining equation

for the exact symmetries of the unperturbed equation , we can get

Then, 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 yields

Solving 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

Table 1: Approximate commutators of approximate symmetry algebra of (1) (, , , ).

0

0
0

0
0

0
0

0
0
0

0

0
0

0
0

0
0
0

0
0
0
0
0
0

0
0
0
0
0

Table 2: Approximate commutators of approximate symmetry algebra of (1) (, ).

0

0
0

0
0

0
0

0
0
0

0

0
0

0
0

0
0
0

0
0
0
0
0
0

0
0
0
0
0

Table 3: Approximate commutators of approximate symmetry algebra of (1) ( , ).

0

0
0

0
0

0
0

0
0
0

0

0
0

0
0

0
0
0

0
0
0
0
0
0

0
0
0
0
0

3.2.2. Approximate Symmetries: The Case

When , the equation admits seven-dimensional approximate Lie algebra as follows:

3.2.3. Approximate Symmetries and Optimal System: The Case

When , the equation admits seven-dimensional approximate Lie algebra as follows:

It is worth nothing that the seven-dimensional approximate Lie algebra is solvable and its finite sequence of ideals is as follows:

In the following, we will construct the optimal system of above Lie algebra . The method used here for obtaining the one-dimensional optimal system of subalgebras is that given in [7]. This approach is taking a general element from the Lie algebra and reducing it to its simplest equivalent form by applying carefully chosen adjoint transformations that are defined as follows.

Definition 6 (see [7]). Let be a Lie group. An optimal system of s-parameter subgroups is a list of conjugacy inequivalent s-parameter subalgebras with the property that any other subgroup is conjugate to precisely one subgroup in the list. Similarly, a list of s-parameter subalgebras forms an optimal system if every s-parameter subalgebra of is equivalent to a unique member of the list under some element of the adjoint representation: , .

Theorem 7 (see [7]). Let and be connected s-dimensional Lie subgroups of the Lie group with corresponding Lie subalgebras and of the Lie algebra of . Then are conjugate subgroups if and only if are conjugate subalgebras.

By Theorem 7, the problem of finding an optimal system of subgroups is equivalent to that of finding an optimal system of subalgebras. To compute the adjoint representation, we use the Lie series

where is the commutator for the Lie algebra and is a parameter,and . In this manner, we construct Table 4 with the th entry indicating .

Table 4: Adjoint representation of approximate symmetry of (1).

Theorem 8. An optimal system of one-dimensional approximate symmetry algebra (case ) of equation (1) is provided by

Proof. Considering the approximate symmetry algebra g of (1), whose adjoint representation was determined in the Table 4, our task is to simplify as many of the coefficients as possible through judicious applications of adjoint maps to , so that is equivalent under the adjoint representation.
Given a nonzero vector

First, suppose that . Scaling if necessary, we can assume that . As for the 7th column of the Table 4, we have

The remaining approximate one-dimensional subalgebras are spanned by vectors of the above form with . If , we have

Next, we act on to cancel the coefficients of and as follows:

If and , the nonzero vector

is equivalent to

If and , we scale to make and then

is equivalent to under the adjoint representation

If and , in the same way as before, the nonzero vector

can be simplified as

If and , we can act

by , to cancel the coefficient of , leading to

The last remaining case occurs when and , for which our earlier simplifications were unnecessary, because the only remaining vectors are the multiples of , on which the adjoint representation acts trivially.

4. Approximately Invariant Solutions

In this section we use two different techniques to construct new approximate solutions of (1) when .

4.1. Approximately Invariant Solutions I

In the beginning of this section we compute an approximately invariant solution based on the The approximate invariants for are determined by

Equivalently

The first equation has two functionally independent solutions and The simplest solutions of the second equation are, respectively, and Therefore, we have two independent invariants to and with respect to

Letting , we obtain for the approximately invariant solutions.

Therefore, we can obtain

Putting (42) into (1), we obtain

where and are arbitrary constants. Hence, we obtained the approximately invariant solution to (1)

In this manner, we compute approximate invariants with respect to the generators of Lie algebra and optimal system, as shown in Table 5.

Table 5: Approximate invariants with respect to the generators of Lie algebra and optimal system.

Operator
Approximate Invariants

,

,

,

,

,

,

,

,

,

,

,

,

4.2. Approximately Invariant Solution II

Now, we apply a different technique to find approximately invariant solutions for (1). We will begin with one exact solution

of the unperturbed equation . Here, is an invariant of the group with the generator

The function given by (45) is invariant under the operator (46).

Using the generators , admitted by , we will take the approximate symmetry

and use it looking for the approximately invariant solution of (1) in the form

Then the invariant equation test can be written as

Note that does not contain the differentiation in ; therefore (49) becomes

whence we obtain the following differential for :

Because , so (51) can be integrated by the variables

Then, denoting by the derivative of with respect to , we have

and (51) becomes

The integration yields

Returning to the variables , , we have

Inserting this in (48) and substituting it into (1) we obtain

Thus, the approximate symmetry (47) provides the following approximately invariant solution (approximate travelling wave):

5. Approximate Conservation Laws

Approximate Lie symmetry can be used to construct the approximate conservation laws, but in this section, we will use partial Lagrangians to construct approximate conservation laws of (1); this is a more concise method.

Definition 9 (see [10]). An operator is a kth-order approximate Lie-Bäcklund symmetry:

where

and

where and the additional coefficients are

and is the Lie characteristic function defined by

Definition 10 (see [10]). The approximate Noether operator associated with an approximate Lie-Bäcklund symmetry operator is given by

where

where is the total derivative operator defined as

and here are Noether operators and the Euler-Lagrange operators are

The Euler-Lagrange, approximate Lie-Bäcklund, and approximate Noether operators are connected by the operator identity

Definition 11 (see [10]). If there exists a function and nonzero functions such that (1) which can be written as , where is an invertible matrix, then, provided that , some is called a partial Lagrangian; otherwise it is the standard Lagrangian. We term differential equations of the form

approximate Euler-Lagrange-type equations.

Definition 12 (see [5]). An approximate Lie-Bäcklund symmetry operator is called an approximate Noether-type symmetry operator corresponding to a partial Lagrangian if and only if there exists a vector defined by

such that

where of is also the characteristic of the conservation law , where

of the approximate Euler-Lagrange-type (68).

Because (1) does not have a Lagrange function and if we put a transform , then (1) becomes

In order to write convenient, we can get

Obviously,

and the Lagrange function is

And the approximate equation is

So the approximate Noether symmetry operator is for

where

So (58) becomes

Put into (82), and let . We obtain

In the following we will consider three cases of and .

First Case. and

where and are arbitrary function for and are differential functions and and

Thus the equation has the following approximate Noether symmetric operators

So the conservation vectors are

Second Case. and

where and are arbitrary function for and are differential functions and

Thus the equation has the following approximate Noether symmetry operators

So the conservation vectors are

Third Case. and ; the conservation laws are the same as the first case.

Data Availability

All data included in this study are available upon request by contact with the corresponding author.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported in part by the Natural Science Foundation of Inner Mongolia of China under grant 2016MS0116.

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