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

Dipartimento di Matematica e Fisica “Ennio de Giorgi”, Università del Salento and INFN, Sezione di Lecce, 73100 Lecce, Italy

E-mail Address: konopel@le.infn.it

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and

G. Ortenzi

http://orcid.org/0000-0003-2192-6737

Dipartimento di Matematica Pura e Applicazioni, Università di Milano Bicocca, 20125 Milano, Italy

E-mail Address: giovanni.ortenzi@unimib.it

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https://doi.org/10.1142/S0129055X20500063Cited by:8 (Source: Crossref)

Abstract

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 $k = 2 n + 1$ , $n = 1, 2, 3, \dots .$ Regularization of these singularities by deformation of plane into plane mappings into surface $S^{2 + k} (\subset ℝ^{2 + k})$ 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.

To the memory of B. A. Dubrovin

Keywords:

AMSC: 58K05, 58K35, 35K59

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