Noether symmetries, conservation laws and Painleve analysis for a Cargo-LeRoux model with the Chaplygin gas in Lagrange variables

We express the Cargo-LeRoux model of fluid dynamics for the description of nonlinear phenomena of fluid dynamics with a Chaplygin gas in terms of Lagrangian variables. The original hyperbolic system of three nonlinear partial differential equations, initially expressed in Eulerian variables, is now reformulated as a nonlinear second-order differential equation. For this equation, we apply symmetry analysis to determine the generator vectors for the infinitesimal transformations that leave the equation invariant. Furthermore, we observe that the equation can be derived from a variational principle, allowing us to apply Noether’s theorem to identify the variational symmetries and corresponding Noetherian conservation laws. Our findings indicate that the number of symmetries and Noetherian conservation laws is independent of the index parameter of the Chaplygin gas. However, the algebraic properties of the admitted Lie symmetries depend on the nature of the fluid source. We demonstrate the application of this analysis by using the symmetry vectors to reduce the equation and determine similarity solutions. Finally, we show that the nonlinear Cargo-LeRoux equations possess the Painlevé property and express the solution in terms of a Laurent expansion.