Discrete Dynamics in Nature and Society
Volume 2019 (2019), Article ID 9609357, 9 pages
https://doi.org/10.1155/2019/9609357

Research Article

Invariant Solutions and Nonlinear Self-Adjointness of the Two-Component Chaplygin Gas Equation

Ben Gao and Yanxia Wang

College of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China

Correspondence should be addressed to Ben Gao; benzi0116@163.com

Received 10 May 2019; Accepted 24 June 2019; Published 4 July 2019

Academic Editor: Xiaohua Ding

Copyright © 2019 Ben Gao and Yanxia Wang. 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.

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]

where

and denote the total derivatives, e.g.,

Substituting and into (2), we acquire the following system of overdetermined equations:

Solving (8), we get

where , , , , , 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.

Table 1: Table of Lie brackets.

0

0
0

0
0
0

0

0
0
0

0
0

0

0

0
0

0

0
0
0

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 as

For , the Lie equations which have parameters with the initial condition are expressed as

The 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 as

We obtain the following representatives:

(1.1.2)
We get the following representatives:

(1.2)
Vector (18) is simplified as

By taking in the transformation , we have . Vector (23) is simplified as

As 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 as

We 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 as

We 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 as

We get the representative

(2.2.2.1)
We obtain vector (32) of the form

We get the representative

Case 2 (). By taking and in the transformations and , we have . Vector (16) is simplified as

Then, 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 variables

and the group-invariant solution is , , i.e.,

Using (41) in (2), we have

where , . 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.

Table 2: The reductions of the two-component Chaplygin gas equation.

Generators
Similarity variables
Reduced equations

, 
, 

, 

, 
,

, 

, 
,

, 

, 
,

, 

, 
,

, 

, 
,

, 

Table 3: Exact solutions of the two-component Chaplygin gas equation.

Generators
Exact solutions

, 

, 

, 

, 

, 

, 

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 PDEs

where , , , , and denotes the collection of all th-order partial derivatives of with respect to .

The adjoint equations of (45) are defined by

where , is the formal Lagrangian, and is the Euler-Lagrange operator written as

where 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 that

where is a particular function.

The following theorem brings about conservation laws [13].

Theorem 3. Any infinitesimal symmetry

admitted by (45) gives rise to a conservation law , where is

where . 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 substitutions

where is an arbitrary constant.

Proof. Let the formal Lagrangian of (2) be of the form

where are two new dependent variables.
Using the equivalent formula (49), the identities

are 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 are

where 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 are

where , , + =

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.

Table 4: Conservation laws of the two-component Chaplygin gas equation.

Generators

0
0

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).

References

C. H. Gu, H. S. Hu, and Z. X. Zhou, Soliton Theory and Its Application , Springer, Berlin, Germany, 1995.

R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, New York, NY, USA, 2004.

V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin, Gemany, 1991.

F. Cariello and M. Tabor, “Painlev expansions for nonintegrable evolution equations,” Physica D: Nonlinear Phenomena, vol. 39, no. 1, pp. 77–94, 1989.

P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2nd edition, 1993.

G. W. Bluman and S. Kumei, Symmetries and Differential Equations, vol. 81 of Applied Mathematical Sciences, Springer-Verlag, Berlin, Germany, 1989.

B. Gao and Y. Zhang, “Symmetries and conservation laws of the Yao–Zeng two-component short-pulse equation,” Boundary Value Problems, vol. 2019, no. 1, 2019.

B. Gao and Y. Zhang, “Symmetry analysis of the time fractional Gaudrey–Dodd–Gibbon equation,” Physica A: Statistical Mechanics and its Applications, vol. 525, pp. 1058–1062, 2019.

C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the Korteweg-deVries equation,” Physical Review Letters, vol. 19, no. 19, pp. 1095–1097, 1967.

G. Badin and F. Crisciani, Variational Formulation of Fluid and Geophysical Fluid Dynamics, Springer, Cham, Swizerland, 2018.

J.-H. He, “Hamiltonian approach to nonlinear oscillators,” Physics Letters A, vol. 374, no. 23, pp. 2312–2314, 2010.

N. H. Ibragimov, “Nonlinear self-adjointness and conservation laws,” Journal of Physics A: Mathematical and Theoretical, vol. 44, no. 43, Article ID 432002, 2011.

N. H. Ibragimov, “A new conservation theorem,” Journal of Mathematical Analysis and Applications, vol. 333, no. 1, pp. 311–328, 2007.

A. Sjoberg, “On double reductions from symmetries and conservation laws,” Nonlinear Analysis: Real World Applications, vol. 10, no. 6, pp. 3472–3477, 2009.

B. Muatjetjeja and C. M. Khalique, “Lie group classification for a generalised coupled lane-emden system in dimension one,” East Asian Journal on Applied Mathematics, vol. 4, no. 4, pp. 301–311, 2014.

V. A. Galaktionov and S. R. Svirshchevskii, Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, Chapman & Hall/CRC, London, UK, 2007.

M. R. Setare, “Holographic Chaplygin gas model,” Physics Letters B, vol. 648, no. 5-6, pp. 329–332, 2007.

M. R. Setare, “Interacting holographic generalized Chaplygin gas model,” Physics Letters B, vol. 654, pp. 1–6, 2007.

J. Liu, T. Zhou, and H. Yang, “Interaction of waves involving delta-shocks in the Chaplygin gas equations,” Southeast Asian Bulletin of Mathematics, vol. 38, no. 6, pp. 825–846, 2014.

Y. Brenier, “Solutions with concentration to the Riemann problem for the one-dimensional Chaplygin gas equations,” Journal of Mathematical Fluid Mechanics, vol. 7, suppl. 3, pp. S326–S331, 2005.

V. Gorini, A. Y. Kamenshchik, U. Moschella, O. F. Piattella, and A. A. Starobinsky, “Gauge-invariant analysis of perturbations in Chaplygin gas unified models of dark matter and dark energy,” Journal of Cosmology and Astroparticle Physics, vol. 2008, no. 2, 2008.

J. S. Alcaniz and J. A. S. Lima, “Measuring the chaplygin gas equation of state from angular and luminosity distances,” The Astrophysical Journal, vol. 618, no. 1, pp. 16–22, 2005.

Y. N. Grigoriev, N. H. Ibragimov, V. F. Kovalev, and S. V. Meleshko, Symmetries of Integro-Differential Equations, vol. 806 of Lecture Notes in Physics, Springer, New York, NY, USA, 2010.