Abelian functions associated with a cyclic tetragonal curve of genus six

TL;DR

This paper develops the theory of Abelian functions for a specific cyclic tetragonal curve of genus six, generalizing classical functions and demonstrating their application to solving the KP-equation.

Contribution

It introduces a new framework for Abelian functions associated with a cyclic tetragonal curve of genus six, including their construction, properties, and applications to integrable systems.

Findings

Constructed Abelian functions using multivariate sigma-functions.

Provided solutions to the KP-equation using these functions.

Derived partial differential equations and addition formulas for the functions.

Abstract

We develop the theory of Abelian functions defined using a tetragonal curve of genus six, discussing in detail the cyclic curve $y^4 = x^5 + \lambda_4x^4 + \lambda_3x^3 + \lambda_2x^2 + \lambda_1x + \lambda_0$. We construct Abelian functions using the multivariate $\sigma$-function associated to the curve, generalising the theory of the Weierstrass $\wp$-function. We demonstrate that such functions can give a solution to the KP-equation, outlining how a general class of solutions could be generated using a wider class of curves. We also present the associated partial differential equations satisfied by the functions, the solution of the Jacobi Inversion Problem, a power series expansion for $\sigma(\bu)$ and a new addition formula.