### 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} + λ_{4}x^{4} + λ_{3}x ^{3} + λ_{2}x^{2} + λ_{1}x + λ_{0}. We construct Abelian functions using the multivariate σ-function associated with the curve, generalizing the theory of the Weierstrass -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 σ(u) and a new addition formula.

Original language | English |
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Article number | 095210 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 42 |

Issue number | 9 |

DOIs | |

Publication status | Published - 2009 |

Externally published | Yes |

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### Bibliographical note

This is an author-created, un-copyedited version of an article acceptedfor publication in Journal of Physics A: Mathematical and Theoretical. The publisher is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://dx.doi.org/10.1088/1751-8113/42/9/095210

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Modelling and Simulation
- Statistics and Probability

### Cite this

*Journal of Physics A: Mathematical and Theoretical*,

*42*(9), [095210]. https://doi.org/10.1088/1751-8113/42/9/095210