Regularization of Hele–Shaw flows, multiscaling expansions and the Painlevé-I equation

Critical processes of ideal integrable models of Hele–Shaw flows are considered. A regularization method based on multiscaling expansions of solutions of the KdV and Toda hierarchies characterized by string equations is proposed. Examples are exhibited in which the tritronquée solution of the Painlevé-I equation turns out to provide the leading term of the regularization.