Darren Crowdy
(Imperial College London, UK)
Reductions of the Benney hierarchy and the Dirichlet problem in two dimensions
Recent research has shown that there are intimate mathematical connections between Laplacian growth problems
of planar domains (such as the evolution of blobs of fluid in a Hele-Shaw cell), the planar Dirichlet problem and its associated Green's function and dispersionless integrable hierarchies (such as the dispersionless Toda and Whitham hierarchies).
In this talk, we show that the connection between the planar Dirichlet problem and integrability extends to the case of the hierarchy of Benney moment equations. It will be shown how a special class of genus-N reductions (due to Gibbons and Tsarev) can be parametrized, in a natural way, using modified Green's functions associated with reflectionally-symmetric multiply connected planar domains.