Dynamics of the ultra-discrete Toda lattice via Pitman's transformation (Mathematical structures of integrable systems and their applications)

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Abstract

By encoding configurations of the ultra-discrete Toda lattice by piecewise linear paths whose gradient alternates between -1 and 1, we show that the dynamics of the system can be described in terms of a shifted version of Pitman's transformation (that is, reflection in the past maximum of the path encoding). This characterisation of the dynamics applies to finite configurations in both the non-periodic and periodic cases, and also admits an extension to infinite configurations. The latter point is important in the study of invariant measures for the ultra-discrete Toda lattice, which is pursued in the parallel work [3]. We also describe a generalisation of the result to a continuous version of the box-ball system, whose states are described by continuous functions whose gradient may take values other than ±1.

"Mathematical structures of integrable systems and their applications". September 5-7, 2018. edited by Shinsuke Iwao. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

Journal

  • RIMS Kokyuroku Bessatsu

    RIMS Kokyuroku Bessatsu B78 235-250, 2020-04

    Research Institute for Mathematical Sciences, Kyoto University

Keywords

Details 詳細情報について

  • CRID
    1050850634912190464
  • NII Article ID
    120006950553
  • NII Book ID
    AA12196120
  • ISSN
    18816193
  • HANDLE
    2433/260639
  • Text Lang
    en
  • Article Type
    departmental bulletin paper
  • Data Source
    • IRDB
    • CiNii Articles

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