### 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 |
---|---|

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

**Abelian functions associated with a cyclic tetragonal curve of genus six.** / England, M.; Eilbeck, J. C.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and Theoretical*, vol. 42, no. 9, 095210. https://doi.org/10.1088/1751-8113/42/9/095210

}

TY - JOUR

T1 - Abelian functions associated with a cyclic tetragonal curve of genus six

AU - England, M.

AU - Eilbeck, J. C.

N1 - This is an author-created, un-copyedited version of an article accepted for 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

PY - 2009

Y1 - 2009

N2 - We develop the theory of Abelian functions defined using a tetragonal curve of genus six, discussing in detail the cyclic curve y4 = x 5 + λ4x4 + λ3x 3 + λ2x2 + λ1x + λ0. 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.

AB - We develop the theory of Abelian functions defined using a tetragonal curve of genus six, discussing in detail the cyclic curve y4 = x 5 + λ4x4 + λ3x 3 + λ2x2 + λ1x + λ0. 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.

U2 - 10.1088/1751-8113/42/9/095210

DO - 10.1088/1751-8113/42/9/095210

M3 - Article

VL - 42

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 9

M1 - 095210

ER -