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A delay analogue of the box and ball system arising from the ultra-discretization of the delay discrete Lotka-Volterra equation

Kenta Nakata, Kanta Negishi, Hiroshi Matsuoka, Ken-ichi Maruno

Abstract

A delay analogue of the box and ball system (BBS) is presented. This new soliton cellular automaton is constructed by the ultra-discretization of the delay discrete Lotka-Volterra equation, which is an integrable delay analogue of the discrete Lotka-Volterra equation. Soliton patterns generated by this delay BBS are classified into normal solitons and abnormal solitons. Normal solitons have a clear relationship to the solitons of the BBS with K kinds of balls. On the other hand, abnormal solitons show various types of novel soliton patterns, which have not been observed in almost all known BBSs. We obtain them by numerical experiments, and then construct τ-functions of them analytically in 1-soliton cases.

A delay analogue of the box and ball system arising from the ultra-discretization of the delay discrete Lotka-Volterra equation

Abstract

A delay analogue of the box and ball system (BBS) is presented. This new soliton cellular automaton is constructed by the ultra-discretization of the delay discrete Lotka-Volterra equation, which is an integrable delay analogue of the discrete Lotka-Volterra equation. Soliton patterns generated by this delay BBS are classified into normal solitons and abnormal solitons. Normal solitons have a clear relationship to the solitons of the BBS with K kinds of balls. On the other hand, abnormal solitons show various types of novel soliton patterns, which have not been observed in almost all known BBSs. We obtain them by numerical experiments, and then construct τ-functions of them analytically in 1-soliton cases.
Paper Structure (15 sections, 8 theorems, 74 equations, 22 figures)

This paper contains 15 sections, 8 theorems, 74 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Construction of a delay box and ball system
  3. Observation of soliton patterns of a delay box and ball system
  4. Normal solitons
  5. Abnormal solitons
  6. The jumping pattern
  7. The long-period pattern
  8. The rule for the time evolution
  9. Normal solitons for $\beta=0$
  10. Normal solitons for general $\beta$
  11. Interaction of normal and jumping solitons
  12. Easy cases
  13. Complicated cases
  14. The relationship to the BBS with $K$ kinds of balls
  15. Conclusions

Key Result

Theorem 3.2

For $\alpha\geq0$ and $\beta\geq1$, the function $J_{n}^{t}$ is defined as follows: where $a,b$ are constant integers, and $C$ is a real constant, and $\left\lfloor{x}\right\rfloor$ is the floor function, i.e., the maximum integer no more than $x$. Then $F_{n}^{t}=J_{n}^{t}$ is a solution of the bilinear equation of the delay BBS (dlyudishlv_bl).

Figures (22)

  • Figure 1: An example of a normal initial state for $\alpha=2,\ \beta=2,\ N=2$.
  • Figure 2: Solitons obtained by normal initial states ($\alpha=1,\ \beta=0,\ N=2$).
  • Figure 3: Solitons obtained by normal initial states ($\alpha=1,\ \beta=1,\ N=2$).
  • Figure 4: Solitons obtained by normal initial states ($\alpha=2,\ \beta=0,\ N=2$).
  • Figure 5: Solitons obtained by normal initial states ($\alpha=2,\ \beta=2,\ N=2$).
  • ...and 17 more figures

Theorems & Definitions (20)

  • Remark 2.1
  • Remark 2.2
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Theorem 3.5
  • proof
  • ...and 10 more