Viscous shocks in Hele–Shaw flow and Stokes phenomena of the Painlevé I transcendent

In Hele–Shaw flows at vanishing surface tension, the boundary of a viscous fluid develops cusp-like singularities. In recent papers Lee et al. (2009, 2008) 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 Painlevé I equation. We argue that the Painlevé 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 Painlevélinear problem.