Unfolding of singularities and differential equations

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.