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<p>We consider the dispersionless limit of the recently introduced multi-component Pfaff–Toda hierarchy. Its dispersionless version is a set of nonlinear differential equations for the dispersionless limit of logarithm of the tau-function (the <i>F</i>-function). They are obtained as limiting cases of bilinear equations of the Hirota–Miwa type. The analysis of the Pfaff–Toda hierarchy is substantially simplified by using the observation that the full (not only dispersionless) <i>N</i>-component Pfaff–Toda hierarchy is actually equivalent to the 2<i>N</i>-component DKP hierarchy. In the dispersionless limit, there is an elliptic curve built in the structure of the hierarchy, with the elliptic modular parameter being a dynamical variable. This curve can be uniformized by elliptic functions, and in the elliptic parametrization the hierarchy acquires a compact and especially nice form.</p>

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Dispersionless version of multi-component Pfaff–Toda hierarchy

  • 期刊论文

  • 发表日期：

    2026年4月20日

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  • A. Savchenko,
  • A. Zabrodin

摘要

We consider the dispersionless limit of the recently introduced multi-component Pfaff–Toda hierarchy. Its dispersionless version is a set of nonlinear differential equations for the dispersionless limit of logarithm of the tau-function (the F-function). They are obtained as limiting cases of bilinear equations of the Hirota–Miwa type. The analysis of the Pfaff–Toda hierarchy is substantially simplified by using the observation that the full (not only dispersionless) N-component Pfaff–Toda hierarchy is actually equivalent to the 2N-component DKP hierarchy. In the dispersionless limit, there is an elliptic curve built in the structure of the hierarchy, with the elliptic modular parameter being a dynamical variable. This curve can be uniformized by elliptic functions, and in the elliptic parametrization the hierarchy acquires a compact and especially nice form.