A genus six cyclic tetragonal reduction of the Benney equations

TL;DR

This paper develops a novel reduction of Benney's equations using genus six cyclic tetragonal curves, explicitly constructing the mapping function as a rational expression involving Kleinian sigma-functions.

Contribution

It introduces a new reduction method for Benney's equations based on complex algebraic curves and explicitly constructs the associated Abelian integral.

Findings

Explicit construction of the mapping function as a rational expression

Connection between Benney's equations and genus six cyclic tetragonal curves

Advancement in algebraic-geometric methods for integrable systems

Abstract

A reduction of Benney's equations is constructed corresponding to Schwartz-Christoffel maps associated with a family of genus six cyclic tetragonal curves. The mapping function, a second kind Abelian integral on the associated Riemann surface, is constructed explicitly as a rational expression in derivatives of the Kleinian sigma-function of the curve.