Lie Symmetries and Similarity Solutions for Rotating Shallow Water

Abstract

Abstract 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 {2A1⊕s\u20052A1}⊕sA1$\\left\\{{2{A_{1}}{\\ \\oplus_{s}}\\ 2{A_{1}}}\\right\\}{\\ \\oplus_{s}}\\ {A_{1}}$ Lie algebra for γ ≠ 1 and 2A1⊕s\u20053A1$2{A_{1}}{\\ \\oplus_{s}}\\ 3{A_{1}}$ 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.