Conservation laws and exact solutions of the Whitham-type equations

• We find the conservation laws for three partial differential equations F–W, modified F–W and Whitham-type equations.
• We analyze the symmetry action of the Lie point symmetry generators on the multipliers.
• By applying symmetry group method to the Whitham-type equation, exact solutions for Whitham-type equation are obtained.

In this paper, we study conservation laws for some partial differential equations. It is shown that interesting conserved quantities arise from multipliers by using homotopy operator that is a powerful algorithmic tool. Furthermore, the invariance properties of the conserved flows with respect to the Lie point symmetry generators are investigated via the symmetry action on the multipliers. Furthermore, the similarity reductions and some exact solutions are provided.