Unfolding of singularities and differential equations

TL;DR

This paper explores the deep connection between singularity theory and differential equations, showing how versal deformations relate to Hamilton-Jacobi systems and integrable hydrodynamic equations, revealing new insights into singularity normal forms.

Contribution

It demonstrates that versal deformations of A,D,E singularities are described by Hamilton-Jacobi equations and links nonversal unfoldings to integrable hydrodynamic systems, highlighting the role of Euler-Poisson-Darboux equations.

Findings

Versal deformations correspond to Hamilton-Jacobi systems.

Nonversal unfoldings relate to integrable hydrodynamic equations.

Generating functions encode higher-order singularity normal forms.

Abstract

Interrelation between Thom's catastrophes and differential equations revisited. It is shown that versal deformations of critical points for singularities of A,D,E type are described by the systems of Hamilton-Jacobi type equations. For particular nonversal unfoldings the corresponding equations are equivalent to the integrable two-component hydrodynamic type systems like classical shallow water equation and dispersionless Toda system and others. Pecularity of such integrable systems is that the generating functions for corresponding hierarchies, which obey Euler-Poisson-Darboux equation, contain information about normal forms of higher order and higher corank singularities.