Nonlinear Differential Equations of Fluid Mechanics: Symmetries, Integrability, Singularities

Book chapter, 2026

In this paper, we illustrate how the ideas from contact and symplectic geometry, as well as from the geometric theory of partial differential equations, can be applied to finding and investigating singular properties of solutions to fundamental equations of fluid dynamics. We give a geometric description of barotropic Euler’s equations on curves and discuss their integrability both in the Jacobi-type systems setting and via differential invariants and quotient PDEs. New classes of multivalued solutions are found, and the description of caustics and shock wave fronts is given. The appearance of phase transitions along the flow in the case of a van der Waals gas is discussed. For barotropic Euler’s equations in 2D and 3D, we find explicit multivalued solutions and discuss singularities of their projections to the space of independent variables. Finally, we address fully compressible Euler’s equations by a similar approach.