Symmetry reductions and exact solutions to the ill-posed Boussinesq equation

• We use the Lie symmetry analysis method to study the ill-posed Boussinesq equation.
• The similarity reductions and exact solutions for this equation are obtained.
• The exact analytic solutions are considered by the power series method.

In this paper, using the Lie symmetry analysis method, we study the ill-posed Boussinesq equation which arises in shallow water waves and non-linear lattices. The similarity reductions and exact solutions for the equation are obtained. Then the exact analytic solutions are considered by the power series method, and the physical significance of the solutions is considered from the transformation group point of view.