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
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 serieswhere is the commutator for the Lie algebra and is a parameter,and . In this manner, we construct Table 4 with the th entry indicating .
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 haveThe 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 EquivalentlyThe 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 obtainPutting (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.
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 solutionof the unperturbed equation . Here, is an invariant of the group with the generatorThe function given by (45) is invariant under the operator (46).
Using the generators , admitted by , we will take the approximate symmetryand use it looking for the approximately invariant solution of (1) in the form
Then the invariant equation test can be written asNote 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:whereandwhere and the additional coefficients areand 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 formapproximate 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 bysuch thatwhere of is also the characteristic of the conservation law , whereof 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 whereSo (58) becomesPut 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.