### 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} + ?_{4}x^{4} + ?_{3}x ^{3} + ?_{2}x^{2} + ?_{1}x + ?_{0}. We construct Abelian functions using the multivariate s-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 s(u) and a new addition formula. © 2009 IOP Publishing Ltd.

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 |