Benney’s equations admit “reductions” to systems of dispersionless pdes with only finitely many N dependent variables. These reductions are parametrised by conformal mappings from the half-plane to the half-plane minus N slits. A large family of these can be constructed as Schwartz-Christoffel mappings; an important subclass of these reduce to integrals of a second kind differential on an algebraic curve. Particular examples of these have been constructed; in particular there are two different constructions for the case of a hyperelliptic curve. Other particular examples have been worked out, and the overall structure is similar to the hyperelliptic case, though there are differences of detail. It is thus important to find the general structure of these, independently of the particular curve being studied. We use the Fay Prime Form to investigate this, and sketch how this approach could be applied to general cyclic (n, s) curves.
Abelian functions prime form algebraic curves Schwartz-Christoffel mapping dispersionless integrable systems algebraic computation
14H40 14H42 14H70 14H51 33F10
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