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 $\gamma \ne 1$ 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.

13 pages, 2 figures, to appear in Zeitschrift für Naturforschung A

References in corpus (2)

• A direct algorithm of one-dimensional optimal system for the group invariant solutions
• Two dimensional dynamical systems which admit Lie and Noether symmetries

Cited by in corpus (5)

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• Invariant conservative difference schemes for shallow water equations in Eulerian and Lagrangian coordinates
• Lie symmetry analysis and one-dimensional optimal system for the generalized 2+1 Kadomtsev-Petviashvili equation
• Shallow-water equations with complete Coriolis force: Group Properties and Similarity Solutions
• Group properties and solutions for the 1D Hall MHD system in the cold plasma approximation