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.

Konopelchenko, B., Ortenzi, G. (2020). On the plane into plane mappings of hydrodynamic type. Parabolic case. REVIEWS IN MATHEMATICAL PHYSICS, 32(3) [10.1142/S0129055X20500063].

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

Ortenzi G.
2020

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 = 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.
Articolo in rivista - Articolo scientifico
Mappings; parabolic systems; singularities
English
2020
32
3
2050006
none
Konopelchenko, B., Ortenzi, G. (2020). On the plane into plane mappings of hydrodynamic type. Parabolic case. REVIEWS IN MATHEMATICAL PHYSICS, 32(3) [10.1142/S0129055X20500063].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/270599
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