Box and Ball System with Numbered Boxes

Yusaku Yamamoto, Akiko Fukuda, Sonomi Kakizaki, Emiko Ishiwata, Masashi Iwasaki, Yoshimasa Nakamura

Research output: Contribution to journalArticlepeer-review


The box and ball system (BBS) models the dynamics of balls moving among an array of boxes. The simplest BBS is derived from the ultradiscretization of the discrete Toda equation, which is one of the most famous discrete integrable systems. The discrete Toda equation can be extended to two types of discrete hungry Toda (dhToda) equations, one of which is the equation of motion of the BBS with numbered balls (nBBS). In this paper, based on the ultradiscretization of the other type of dhToda equation, we present a new nBBS in which not balls, but boxes, are numbered. We also investigate conserved quantities with respect to balls and boxes, the solitonical nature of ball motions, and a scattering rule in collisions of balls to clarify the characteristics of the resulting nBBS.

Original languageEnglish
Article number13
JournalMathematical Physics Analysis and Geometry
Issue number2
Publication statusPublished - 2022 Jun


  • Box and ball system
  • Discrete hungry Toda equation
  • Numbered boxes
  • Solitonical nature
  • Ultradiscretization

ASJC Scopus subject areas

  • Mathematical Physics
  • Geometry and Topology


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