Abstract
In this paper, the Lie group method is performed on a special dark fluid, the Chaplygin gas, which describes both dark matter and dark energy in the present universe. Based on an optimal system of one-dimensional subalgebras, similarity reductions and group invariant solutions are given. Finally, by means of Ibragimov’s method, conservation laws are obtained.
1. Introduction
Nonlinear partial differential equations (PDEs) play an indispensable role in the nature. A host of natural phenomena is able to be well simulated by them. The multitude of methods have been explored to find exact solutions for the PDEs. Some of the most representative methods are Bäcklund method [1], Hirota bilinear method [2], Darboux transformation [3], Painleve expansion method [4], Lie symmetry analysis [5–8], the inverse scattering method [9], etc. Lie symmetry analysis, a quite effective method among them, can get crucially explicit solutions of PDEs. On the other hand, an important issue regarding the PDEs is to obtain their conservation laws. Conservation laws can be obtained by variational principle or Hamilton’s principle [10, 11]. For the PDEs which do not have a Lagrangian, based on admitted symmetries [12], Ibragimov proposed the concept of the adjoint equation for the study of conservation laws by using conservation law theorem in [13]. Moreover, we can apply conservation laws to obtain exact solutions of PDEs [14, 15].
The aim of this paper is to study symmetries, explicit solutions, nonlinear self-adjointness, and conservation laws of the two-component Chaplygin gas equation [16]:This system consists of the equation of continuity for the density and Euler’s force equation for an ideal fluid of zero vorticity with the velocity potential , where the subscripts stand for the partial derivatives. Equation (1) was introduced by Chaplygin, Tsien, and von Karman as a mathematical approximation for calculating the lifting force on a wing of an airplane in aerodynamics [16]. The two-component Chaplygin gas equation can describe both dark matter and dark energy in the present universe [17, 18]. This system has received extensive attention. For example, the Riemann problem of (1) and the interaction of waves were exhibited in [19]; the Riemann solutions with concentration were derived in [20]; the perturbative behaviour of (1) was analyzed in [21]; qualitative and quantitative aspects of (1) were investigated in [22].
In the end, this paper is arranged as follows. In Section 2, we focus on obtaining Lie point symmetry of (1). Section 3 mainly constructs similarity reductions and group invariant solutions. Conservation laws, using lbragimov’s method, have been set up in Section 4. Some conclusions are presented in the last section.
2. Lie Point Symmetries
In order to use the Lie group method to analyze the model, we assumed that the gas is incompressible. Let the pressure , and we can transform (1) to
In this section, we use Lie point symmetry method for (2) and acquire its infinitesimal generators, commutation table of Lie algebra.
First of all, let us consider an infinitesimal generator admitted by (2):
According to the invariance conditions [5], we have
The prolongations for (2) of the infinitesimal generator (3) are [5]whereand denote the total derivatives, e.g.,Substituting and into (2), we acquire the following system of overdetermined equations:
Solving (8), we getwhere , , , , , and are arbitrary constants. The Lie algebra of infinitesimal symmetry of (2) is spanned by the following generators:
To classify all the group-invariant solutions, we construct an optimal system of one-dimensional subalgebras of (2) by using the method in [23], which only depends on the commutator table. The commutator relations about are listed in Table 1.
An arbitrary operator is of the form
In order to discuss the linear transformations of the vector , the following generator is given as where is determined by . Based on (12) and Table 1, can be represented asFor , the Lie equations which have parameters with the initial condition are expressed asThe transformations of the solutions of these equations are given as follows:The structure of the optimal system needs a simplification of the vector: by using the transformations . We mainly concentrate on finding a simplest representative of vector (16). The structure is classified into two cases.
Case 1 ( = ). Consider vector (16) of the form(1)
By taking in the transformation , we have . Vector (17) is hence reduced as(1.1)
By taking in the transformation , we have . Vector (18) is simplified as(1.1.1)
By taking in the transformation , we have . Vector (19) is simplified asWe obtain the following representatives:(1.1.2)
We get the following representatives:(1.2)
Vector (18) is simplified asBy taking in the transformation , we have . Vector (23) is simplified asAs a result, we get the following representatives:(2)
We obtain vector (17) of the form(2.1)
By taking and in the transformations and , we have . Vector (26) is simplified asWe get the following representatives:(2.2)
We obtain vector (26) of the form(2.2.1)
By taking in the transformation , we have . Vector (29) is simplified asWe get the following representatives:(2.2.2)
We obtain vector (29) of the form(2.2.2.1)
By taking in the transformation , we have . Vector (32) is simplified asWe get the representative(2.2.2.1)
We obtain vector (32) of the formWe get the representative
Case 2 (). By taking and in the transformations and , we have . Vector (16) is simplified asThen, we obtain the following representatives:Hence, by gathering the operators (21), (22), (25), (28), (31), (34), (36), and (38), we get to the following theorem.
Theorem 1. The operators in create an optimal system:
3. Exact Solutions
In this section, we cope with the reductions and derive group invariant solutions of (2).
Case 1 (reduction by ). For generator , we have similarity variablesand the group-invariant solution is , , i.e.,Using (41) in (2), we havewhere , . Therefore, (2) has a solution , , where are arbitrary constants.
Case 2 (reduction by ). Similar to Case 1, we have , , . And the reduced equation is where , . Therefore, (2) has a solution , , where is an arbitrary constant.
Case 3. For generator , we have , , where . The reduced equation is where , . Therefore, (2) has a solution , + , where are arbitrary constants.
Some of the reductions for the optimal system of one-dimensional subalgebras and exact solutions are represented in Tables 2 and 3, respectively.
4. Conservation Laws
In this section, we prove that (2) is nonlinear self-adjoint. Moreover, using Ibragimov’s method, we construct conservation laws.
4.1. Preliminaries
Consider an th-order system of PDEswhere , , , , and denotes the collection of all th-order partial derivatives of with respect to .
The adjoint equations of (45) are defined bywhere , is the formal Lagrangian, and is the Euler-Lagrange operator written aswhere denotes the total derivative with respect to .
Definition 2 (see [12]). System (45) is said to be nonlinearly self-adjoint if the adjoint system (46) is satisfied for all solutions of (45) upon a substitution:such that .
Definition 2 means thatwhere is a particular function.
The following theorem brings about conservation laws [13].
Theorem 3. Any infinitesimal symmetryadmitted by (45) gives rise to a conservation law , where iswhere . The Lagrangian should be represented with respect to all mixed derivatives .
4.2. Nonlinear Self-Adjointness
Following Definition 2 and Theorem 3, we prove that (2) is nonlinear self-adjoint.
Theorem 4. Equation (2) is nonlinearly self-adjoint under the substitutionswhere is an arbitrary constant.
Proof. Let the formal Lagrangian of (2) be of the formwhere are two new dependent variables.
Using the equivalent formula (49), the identitiesare established under the substitutions , .
Equation (54) can be split by using the coefficients of different order derivatives of and , and then we obtain the system in unknown variables , whose solutions arewhere is an arbitrary constant.
4.3. Construction of Conservation Laws
For the generator , by Theorem 3, the conservation law of (2) is expressed in the form , and the conserved vectors arewhere , , + =
By simplifying , , we have
For the generator , we obtain , . On the basis of formulas (57) and (58), we obtain
The remaining conservation laws of other generators for (2) are shown in Table 4.
5. Conclusions
In this paper, we apply Lie group method to the two-component Chaplygin gas equation. Based on this method, the optimal system of one-dimensional subalgebras and similarity reductions are derived. Furthermore, exact solutions of the reduced equations are constructed. Eventually, it is shown that (2) is nonlinearly self-adjoint, and, at the same time, we obtain its conservation laws. Our results can be used to characterize the problems of both dark matter and dark energy in the present universe.
Data Availability
The data used to support the findings of this study are available within the paper.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
This work was supported by the Natural Science Foundation of Shanxi (No. 201801D121018).