2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/101367We study a family of solutions to the Saffman-Taylor problem with zero surface tension at a critical regime. In this regime, the interface develops a thin singular finger. The flow of an isolated finger is given by the Whitham equations for the KdV integrable hierarchy. We show that the flow describing bubble break-off is identical to the Gurevich-Pitaevsky solution for regularization of shock waves in dispersive media. The method provides a scheme for the continuation of the flow through singularites.Some typos corrected, added journal referenceExactly Solvable and Integrable SystemsSoft Condensed MatterPattern Formation and SolitonsFluid Dynamicstext