On optimal system, exact solutions and conservation laws of the Broer-Kaup system

The Broer-Kaup system is an important physical model which is used to model the bi-directional propagation of long waves in shallow water. In this paper, Lie symmetry analysis is performed on the Broer-Kaup system. We get the Lie point symmetries and optimal system of one-dimensional subalgebras. Similarity reductions of the system are obtained based on optimal system of one-dimensional subalgebras. We present some exact solutions of the system, which include similarity solutions and travelling wave solutions. Furthermore, some conservation laws are generated via multipliers. The conservation laws associated with symmetries of this equation are constructed by utilizing the new conservation theorem.