Mark Mineev-Weinstein: New pattern selection results in a Hele-Shaw cell

Pattern formation selection problems (far from equilibrium) was and still is a long-standing challenge for theoretical physics since 1950s, when pioneering experiments were conducted by Sir G.I. Taylor.
The main difficulty was the absence of conventional mathematical tools to deal with unstable environment. Still in 1980s the significant efforts of several theoretical groups produced results on the pattern selection through a surface tension. In these works they used the so-called "Asymptotic beyond all orders", developed by M. Kruskal and H.
Segur, using a subtle WKB-based technique, developed in 1960s by Pokrovsky and Khalatnikov for an electron reflection over a potential barrier.

In this talk I overview previous efforts and will report a new result obtained jointly with G.L. Vacsonselos in a problem of selection of bubbles from a continuous family of admissible solutions without surface tension. The results are in the excellent agreement with experiments.
This was possible to obtain due to a remarkable and powerful integrable structure of the nonlinear interface dynamics equations, related to the dispersionless 2D integrable Toda hierarchy, which we have elaborated earlier.

The challenges still left in the problem will be also discussed.