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A study of the discrete Pv equation: Miura transformations and particular solutions

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Abstract

We derive the form of the Miura transformation of the discrete Pv equation and show that it is indeed an auto-Bäcklund transformation, i.e. it relates the discrete Pv to itself. Using this auto-Bäcklund, we obtain the Schlesinger transformations of discrete Pv which relate the solution for one set of the parameters of the equation to that of another set of neighbouring parameters. Finally, we obtain particular solutions of the discrete Pv (i.e. solutions that exist only for some specific values of the parameters). These solutions are of two types: solutions involving the confluent hypergeometric function (on codimension-one submanifold of parameters) and rational solutions (on codimension-two submanifold of parameters).

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Authors and Affiliations

  1. Department of Mathematics, Pondicherry University, Kalapet, 605104, Pondicherry, India

    K. M. Tamizhmani

  2. CPT, Ecole Polytechnique, CNRS, UPR 14, 91128, Palaiseau, France

    A. Ramani

  3. LPN, Université Paris VII, Tour 24-14, 5ème étage, 75251, Paris, France

    B. Grammaticos

  4. Department of Applied Mathematics, Faculty of Engineering, Hiroshima University, 1-4-1 Kagamiyama, 739, Higashi-Hiroshima, Japan

    Y. Ohta

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  1. K. M. Tamizhmani
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  2. A. Ramani
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  3. B. Grammaticos
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  4. Y. Ohta
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Tamizhmani, K.M., Ramani, A., Grammaticos, B. et al. A study of the discrete Pv equation: Miura transformations and particular solutions. Lett Math Phys 38, 289–296 (1996). https://doi.org/10.1007/BF00398353

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  • DOI: https://doi.org/10.1007/BF00398353

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