Ultradiscrete analogues of the hard-spring equation and its conserved quantity

  • Shin IsojimaEmail author
  • Hirotaka Toyama
Original Paper Area 1


The ultradiscrete analogues with parity variables of the so-called hard spring equation and its conserved quantity are proposed. Solutions of the resulting equation are constructed for many initial values, and a diagram is proposed to illustrate the structure of each solution. The behavior of the solutions is classified into four (or precisely five) types, two of which are periodic. Then, the ultradiscrete analogue of the conserved quantity is investigated to determine whether the conserved quantity is preserved for each solution. Three types of behavior are observed for the “ultradiscretized conserved quantity,” which is actually preserved in one type but not always in the other types. However, perfect matching between the behavior of the ultradiscrete solutions and that of the ultradiscretized conserved quantity is observed, and the mathematical structure partly survives through ultradiscretization.


Duffing equation Ultradiscretization Integrable systems Conserved quantity 

Mathematics Subject Classification

34A34 39A10 39A23 



The authors are grateful to Mr. Yasuhiro Asakawa for his elementary study of the p-ultradiscrete nonlinear oscillator equation in his Bachelor thesis. This research was supported by JSPS KAKENHI 26790082.


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Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Industrial and Systems Engineering, Faculty of Science and EngineeringHosei UniversityKoganei-shiJapan

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