A common integrable structure in the hermitian matrix model and Hele-Shaw flows

It is proved that the system of string equations of the dispersionless 2-Toda hierarchy which arises in the planar limit of the hermitian matrix model also underlies certain processes in Hele-Shaw flows.

💡 Research Summary

The paper establishes that the string equations of the dispersionless 2‑Toda (d2‑Toda) hierarchy, which emerge in the planar (large‑N) limit of the Hermitian matrix model, also underlie the dynamics of bubble break‑off in Hele‑Shaw flows. The authors begin by recalling the d2‑Toda hierarchy formulated in terms of two pairs of Lax‑Orlov functions ((z,m)) and ((\bar z,\bar m)). The string equations they focus on are the simple constraints (z=\bar z) and (m=\bar m). The first constraint forces a reduction to the 1‑Toda hierarchy, yielding the explicit representation \