Classification of nonlocal symmetries and exact solutions for 3 × 3 Chaplygin gas equation with conservation laws

Abstract

In this paper, we study the one-dimensional isentropic compressible Euler system for the Chaplygin gas through Lie symmetry analysis. The one-dimensional optimal subalgebras are classified using the adjoint transformation and the invariant functions. We derived several new exact solutions from the optimal subalgebras and investigated the physical behavior of some solutions graphically. Next, a tree of nonlocally related partial differential equations (PDEs) is presented and we classify the nonlocal symmetry of the given system. Futher, some nontrivial exact solutions for the given model are constructed using nonlocal symmetries. Furthermore, using the traveling wave transformation, which is invariant under the symmetry group, we obtain solutions of the nature of peakon-type and kink-type solitons. Then, conservation laws are constructed through the direct multipliers method. Finally, the evolutionary behavior of a C1-wave is investigated using one of the developed solutions.