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) [8,9] 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.