Discrete Dynamics in Nature and Society

Volume 2019, Article ID 9609357, 9 pages

https://doi.org/10.1155/2019/9609357

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

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

Correspondence should be addressed to Ben Gao; moc.361@6110izneb

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]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.