On the plane into plane mappings of hydrodynamic type. Parabolic case

Singularities of plane into plane mappings described by parabolic two-component systems of quasi-linear partial differential equations of the first order are studied. Impediments arising in the application of the original Whitney's approach to such a case are discussed. Hierarchy of singularities is analyzed by the double-scaling expansion method for the simplest 2-component Jordan system. It is shown that flex is the lowest singularity while higher singularities are given by (k + 1,k + 2) curves which are of cusp type for k = 2n + 1, n = 1, 2, 3,.... Regularization of these singularities by deformation of plane into plane mappings into surface S2+k(⊂ ℝ2+k) to plane is discussed. Applicability of the proposed approach to other parabolic type mappings is noted. We finally compare the results obtained for the parabolic case with non-generic gradient catastrophes for hyperbolic systems.