Nonlocal symmetries classifications and exact solution of Chaplygin gas equations

Abstract

In this work, a complete tree of nonlocally related partial differential equations of Chaplygin gas equations is constructed. This tree includes the systems, which are obtained through local conservation laws and the local symmetry based method. We classify the nonlocal symmetries from potential systems as well as inverse potential systems (IPSs). Furthermore, we propose a systematic algorithm for identification of nonlocal symmetries through IPSs by combining the ideas in the studies of Bluman and Yang [J. Math. Phys. 54, 093504 (2013)] and Yang and Cheviakov [J. Math. Phys. 55, 083514 (2014)]. Finally, we obtain a new exact solution through nonlocal symmetry analysis and physical behavior of solution is presented.