On Generic Singularities of Solutions to the 1D Gas Flow Equations: Chaplygin and Bechert–Stanyukovich Cases

Typical singularities of solutions to the equations of one-dimensional isentropic gas dynamics in the Chaplygin and Bechert–Stanyukovich cases are studied. These two cases violate a condition even stronger than the condition of strong nonlinearity and serve as an approximation to real gases. The singularities of these cases have not been studied before. The section of the cusp singularity ( \(A_{3}\) ) for the Chaplygin case and the section of the hyperbolic umbilic singularity ( \(D_{4+}\) ) for both cases are described. The complete inheritance of the canonical form from the section of the cusp singularity of the solution to the hodograph image of the wave equation in the Chaplygin case is noted. The simplification of the canonical form in the Bechert–Stanyukovich case is noted. The work uses standard methods and results from the theory of singularities of differentiable mappings.