In Hele-Shaw flows at vanishing surface tension, the boundary of a viscous
fluid develops cusp-like singularities. In recent papers [1, 2] we have showed
that singularities trigger viscous shocks propagating through the viscous
fluid. Here we show that the weak solution of the Hele-Shaw problem describing
viscous shocks is equivalent to a semiclassical approximation of a special real
solution of the Painleve I equation. We argue that the Painleve I equation
provides an integrable deformation of the Hele-Shaw problem which describes
flow passing through singularities. In this interpretation shocks appear as
Stokes level-lines of the Painleve linear problem.Comment: A more detailed derivation is include
