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

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]  M. Setare Holographic Chaplygin gas model , 2007, 0704.3679.

[2]  V. Matveev,et al.  Darboux Transformations and Solitons , 1992 .

[3]  Ji-Huan He,et al.  Hamiltonian approach to nonlinear oscillators , 2010 .

[4]  Ben Gao,et al.  Symmetry analysis of the time fractional Gaudrey–Dodd–Gibbon equation , 2019, Physica A: Statistical Mechanics and its Applications.

[5]  Y. Brenier Solutions with Concentration to the Riemann Problem for the One-Dimensional Chaplygin Gas Equations , 2005 .

[6]  Ben Muatjetjeja,et al.  Lie group classification for a generalised coupld lane–emden system in dimension one , 2014 .

[7]  C. S. Gardner,et al.  Method for solving the Korteweg-deVries equation , 1967 .

[8]  M. Tabor,et al.  Painlevé expansions for nonintegrable evolution equations , 1989 .

[9]  A. Sjöberg,et al.  On double reductions from symmetries and conservation laws , 2009 .

[10]  Ben Gao,et al.  Symmetries and conservation laws of the Yao–Zeng two-component short-pulse equation , 2019, Boundary Value Problems.

[11]  M. Setare Interacting holographic generalized Chaplygin gas model , 2007, 0708.0118.

[12]  N. Ibragimov A new conservation theorem , 2007 .

[13]  J. Meinhardt,et al.  Symmetries and differential equations , 1981 .

[14]  Measuring the Chaplygin Gas Equation of State from Angular and Luminosity Distances , 2003, astro-ph/0308465.

[15]  Nail H. Ibragimov,et al.  Nonlinear self-adjointness and conservation laws , 2011, 1107.4877.

[16]  Yurii N. Grigoriev,et al.  Symmetries of Integro-Differential Equations , 2010 .

[17]  C. Gu,et al.  Soliton theory and its applications , 1995 .