Symmetry reductions and exact solutions to the systems of nonlinear partial differential equations**This work was supported by the National Natural Science Foundation of China under Grant No. 11171041 and the high-level personnel foundation of Liaocheng University under grant No. 31805.

Abstract

In this paper, the Lie symmetry analysis is performed on the four systems of nonlinear partial differential equations (PDEs). One is the system of nonlinear coupled PDEs in a real physical application. The other system describes the unsteady liquid and gas flows in long pipelines. The third is the approximate equations for long waves in shallow water. The fourth is the general classical dispersiveless long wave equations. All of the point symmetries of the systems are presented for the first time, then the symmetry reductions and exact solutions to the systems are investigated.