Lie Symmetries and Similarity Solutions for Rotating Shallow Water

We study a nonlinear system of partial differential equations that describe rotating shallow water with an arbitrary constant polytropic index γ 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 { 2 A 1 ⊕ s 2 A 1 } ⊕ s A 1 \2A₁\ ⊕ₛ\ 2A₁\\ ⊕ₛ\ A₁ Lie algebra for γ ≠ 1 and 2 A 1 ⊕ s 3 A 1 2A₁\ ⊕ₛ\ 3A₁ for γ = 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.