Lie Symmetries and Exact Solutions of Shallow Water Equations with Variable Bottom

Abstract In the present paper, Lie symmetries of nonlinear shallow water equations with variable shapes of the bottom that include horizontal, inclined plane and a parabolic bottom are obtained. Exact particular solutions of the governing system are then obtained using the invariance of the system under these symmetries using Lie’s method. The evolutionary behaviour of the C1$${C^1}$$ discontinuity wave, influenced by the amplitude of the discontinuity wave and the geometry of the bottom, is discussed in detail and some contrasting observations are made.