Lie symmetries and similarity solutions for rotating shallow water

Abstract: We study a nonlinear system of partial differential equations which describe rotating shallow water with an arbitrary constant polytropic index $\gamma $ for the fluid. In our analysis we apply the theory of symmetries for differential equations and we determine that the system of our study is invariant under a five dimensional Lie algebra. The admitted Lie symmetries form the $\left{ 2A_{1}\oplus {s}2A{1}\right} \oplus {s}A{1}$ Lie algebra for $\gamma \neq 1$ and $2A_{1}\oplus {s}3A{1}$ for $\gamma =1$. The application of the Lie symmetries is performed with the derivation of the corresponding zero-order Lie invariants which applied to reduce the system of partial differential equations into integrable systems of ordinary differential equations. For all the possible reductions the algebraic or closed-form solutions are presented. Travel-wave and scaling solutions are also determined.