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On the Riemann theta function of a trigonal curve and solutions of the Boussinesq and KP equations

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Abstract

Recently, considerable progress has been made in understanding the nature of the algebro-geometrical superposition principles for the solutions of nonlinear completely integrable evolution equations, and mainly for the equations related to hyperelliptic Riemann surfaces. Here we find such a superposition formula for particular real solutions of the KP and Boussinesq equations related to the nonhyperelliptic curve ω4 = (λ − E 1) (λ − E 2) (λ − E 3) (λ − E 4). It is shown that the associated Riemann theta function may be decomposed into a sum containing two terms, each term being the product of three one-dimensional theta functions. The space and time variables of the KP and Boussinesq equations enter into the arguments of these one-dimensional theta functions in a linear way.

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Author notes

  1. V. B. Matveev

    Present address: Department of Physics, Division of Mathematical Physics, Leningrad University, Pervomayaskaya 100, 198904, Leningrad, USSR

  2. A. O. Smirnov

    Present address: Department of Mathematics of the Leningrad Institute of Aviation Instrumentation, Gertzena 67, 190000, Leningrad, USSR

Authors and Affiliations

  1. II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 2 Hamburg 50, West Germany

    V. B. Matveev & A. O. Smirnov

Authors
  1. V. B. Matveev
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  2. A. O. Smirnov
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On leave from Leningrad State University and Leningrad Institute of Aviation Instrumentation.

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Matveev, V.B., Smirnov, A.O. On the Riemann theta function of a trigonal curve and solutions of the Boussinesq and KP equations. Lett Math Phys 14, 25–31 (1987). https://doi.org/10.1007/BF00403466

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  • DOI: https://doi.org/10.1007/BF00403466

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