Genus-Nalgebraic reductions of the Benney hierarchy within a Schottky model

Abstract

By exploiting a new theoretical connection between reductions of the Benney hierarchy and the Dirichlet problem for Laplace's equation, the solution to a spectral problem associated with N-parameter algebraic reductions of the Benney hierarchy is found explicitly. The solutions can be written in terms of the modified Green's function associated with reflectionally symmetric, N-connected planar domains whose 'holes' are all centred on the symmetry axis. Explicit formulae for the modified Green's function in a canonical class of circular domains are constructed using a Schottky model of the Schottky double of these domains. Uniformizations of the spectral problem associated with two different types of reductions then follow from these formulae.