The Benney hierarchy and the Dirichlet boundary problem in two dimensions

A theoretical connection between reductions of the Benney hierarchy and the Dirichlet problem for Laplace's equation in the plane is made. The connection is used to deduce general formulas for the uniformizations of two spectral functions associated with N-parameter reductions of the hierarchy. Two types of reduction are considered: one type has been considered by previous authors using alternative arguments, the second type is new. The formulas are general and are expressed in terms of the modified Green's function (for Laplace's equation) in arbitrary N-connected, reflectionally-symmetric, planar domains. The Benney moments are found to be purely geometrical quantities associated with these domains.