Soliton Distribution in the Ball and Box Cellular Automation Model

Alpha Soko; James Makungu

doi:doi:10.11648/j.ajmcm.20190401.14

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Soliton Distribution in the Ball and Box Cellular Automation Model

Alpha Soko, James Makungu

Published in American Journal of Mathematical and Computer Modelling (Volume 4, Issue 1)

Received: 3 April 2019 Accepted: 16 May 2019 Published: 13 June 2019

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Abstract

Solitons are very important waves in Mathematical Physics. They help model many physical situations for instance tsunami waves, application in optical communication systems, application in plasma physics and application in laser guided technology among others. Despite enormous applications, solving the non-linear equations which have soliton solutions is challenging be- cause the nonlinear characteristic of the system abruptly changes due to some slight changes of valid parameters including time. A remedy to this is to use cellular automation models to model soliton dynamics. Cellular automation models do not change in response to slight changes on parameters. In this paper, using computer simulation, an investigation of the distribution of the number of solitons from the Ball and Box cellular automation model was examined. The distribution of the number of solitons from the two and three colour Ball and Box cellular automation was established. Using the online integer sequence, for the two colour Ball and Box cellular automation model, it was found that the distribution of the number of the solitons is indexed by the binomial coefficients. On the Other hand, for the three colour Ball and Box cellular automation model, it was found that only solitons of lengths one to four possesses the distribution while the other soliton lengths do not possess any distribution function.

Published in	American Journal of Mathematical and Computer Modelling (Volume 4, Issue 1)
DOI	10.11648/j.ajmcm.20190401.14
Page(s)	27-30
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Solitons, Ball and Box Model, Cellular Automation, Integer Sequence, Binomial Coefficients

References

[1]	D. Takahashi, J. Satsuma, A soliton cellular automaton, Journal of the Physical Society of Japan 59 (10) (1990) 3514–3519.
[2]	T. Tokihiro, D. Takahashi, J. Matsukidaira, J. Satsuma, From soliton equations to integrable cellular automata through a limiting procedure, Physical Review Letters 76 (18) (1996) 3247.
[3]	M. Torii, D. Takahashi, J. Satsuma, Combinatorial representation of invariants of a soliton cellular automaton, Physica D: Nonlinear Phe- nomena 92 (3-4) (1996) 209–220.
[4]	P. Caudrey, Memories of hirota’s method: application to the reduced Maxwell–bloch system in the early 1970s, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 369 (1939) (2011) 1215–1227.
[5]	T. Lam, P. Pylyavskyy, R. Sakamoto, Box-basket-ball systems., Reviews in Mathematical Physics 24 (08) (2012) 1250019.
[6]	S. Iwao, The periodic box-ball system and tropical curves., RIMS Kˆokyuˆroku Bessatsu B 13 (2009) 157–174.
[7]	A. Kuniba, T. Takagi, Bethe ansatz, inverse scattering transform and tropical riemann theta function in a periodic soliton cellular automa- ton for aˆ(1) n, Symmetry, Integrability and Geometry: Methods and Applications 6 (0) (2010) 13–52.
[8]	J. Mada, M. Idzumi, T. Tokihiro, Path description of conserved quanti- ties of generalized periodic box-ball systems., Journal of mathematical physics 46 (2) (2005) 022701.
[9]	K. Fukuda, Y. Yamada, M. Okado, Energy functions in box ball sys- tems., International Journal of Modern Physics A 15 (09) (2000) 1379– 1392.
[10]	N. J. Sloane, et al., The on-line encyclopedia of integer sequences (2003). URL https://oeis.org/.
[11]	A. Kuniba, M. Okado, R. Sakamoto, T. Takagi, Y. Yamada, Crystal interpretation of kerov–kirillov–reshetikhin bijection., Nuclear Physics B 740 (3) (2006) 299–327.
[12]	A. Kuniba, R. Sakamoto, Combinatorial bethe ansatz and generalized periodic box-ball system., Reviews in Mathematical Physics 20 (05) (2008) 493–527.

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Alpha Soko, James Makungu. (2019). Soliton Distribution in the Ball and Box Cellular Automation Model. American Journal of Mathematical and Computer Modelling, 4(1), 27-30. https://doi.org/10.11648/j.ajmcm.20190401.14

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ACS Style

Alpha Soko; James Makungu. Soliton Distribution in the Ball and Box Cellular Automation Model. Am. J. Math. Comput. Model. 2019, 4(1), 27-30. doi: 10.11648/j.ajmcm.20190401.14

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AMA Style

Alpha Soko, James Makungu. Soliton Distribution in the Ball and Box Cellular Automation Model. Am J Math Comput Model. 2019;4(1):27-30. doi: 10.11648/j.ajmcm.20190401.14

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@article{10.11648/j.ajmcm.20190401.14,
  author = {Alpha Soko and James Makungu},
  title = {Soliton Distribution in the Ball and Box Cellular Automation Model},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {4},
  number = {1},
  pages = {27-30},
  doi = {10.11648/j.ajmcm.20190401.14},
  url = {https://doi.org/10.11648/j.ajmcm.20190401.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20190401.14},
  abstract = {Solitons are very important waves in Mathematical Physics. They help model many physical situations for instance tsunami waves, application in optical communication systems, application in plasma physics and application in laser guided technology among others. Despite enormous applications, solving the non-linear equations which have soliton solutions is challenging be- cause the nonlinear characteristic of the system abruptly changes due to some slight changes of valid parameters including time. A remedy to this is to use cellular automation models to model soliton dynamics. Cellular automation models do not change in response to slight changes on parameters. In this paper, using computer simulation, an investigation of the distribution of the number of solitons from the Ball and Box cellular automation model was examined. The distribution of the number of solitons from the two and three colour Ball and Box cellular automation was established. Using the online integer sequence, for the two colour Ball and Box cellular automation model, it was found that the distribution of the number of the solitons is indexed by the binomial coefficients. On the Other hand, for the three colour Ball and Box cellular automation model, it was found that only solitons of lengths one to four possesses the distribution while the other soliton lengths do not possess any distribution function.},
 year = {2019}
}

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TY  - JOUR
T1  - Soliton Distribution in the Ball and Box Cellular Automation Model
AU  - Alpha Soko
AU  - James Makungu
Y1  - 2019/06/13
PY  - 2019
N1  - https://doi.org/10.11648/j.ajmcm.20190401.14
DO  - 10.11648/j.ajmcm.20190401.14
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 27
EP  - 30
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20190401.14
AB  - Solitons are very important waves in Mathematical Physics. They help model many physical situations for instance tsunami waves, application in optical communication systems, application in plasma physics and application in laser guided technology among others. Despite enormous applications, solving the non-linear equations which have soliton solutions is challenging be- cause the nonlinear characteristic of the system abruptly changes due to some slight changes of valid parameters including time. A remedy to this is to use cellular automation models to model soliton dynamics. Cellular automation models do not change in response to slight changes on parameters. In this paper, using computer simulation, an investigation of the distribution of the number of solitons from the Ball and Box cellular automation model was examined. The distribution of the number of solitons from the two and three colour Ball and Box cellular automation was established. Using the online integer sequence, for the two colour Ball and Box cellular automation model, it was found that the distribution of the number of the solitons is indexed by the binomial coefficients. On the Other hand, for the three colour Ball and Box cellular automation model, it was found that only solitons of lengths one to four possesses the distribution while the other soliton lengths do not possess any distribution function.
VL  - 4
IS  - 1
ER  -

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Author Information

Alpha Soko

Department of Mathematics, Mzuzu University, Mzuzu, Malawi
James Makungu

Department of Mathematics, University of Dar es Salaam, Dar es Salaam, Tanzania

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APA Style

Alpha Soko, James Makungu. (2019). Soliton Distribution in the Ball and Box Cellular Automation Model. American Journal of Mathematical and Computer Modelling, 4(1), 27-30. https://doi.org/10.11648/j.ajmcm.20190401.14

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ACS Style

Alpha Soko; James Makungu. Soliton Distribution in the Ball and Box Cellular Automation Model. Am. J. Math. Comput. Model. 2019, 4(1), 27-30. doi: 10.11648/j.ajmcm.20190401.14

Copy | Download

AMA Style

Alpha Soko, James Makungu. Soliton Distribution in the Ball and Box Cellular Automation Model. Am J Math Comput Model. 2019;4(1):27-30. doi: 10.11648/j.ajmcm.20190401.14

Copy | Download

@article{10.11648/j.ajmcm.20190401.14,
  author = {Alpha Soko and James Makungu},
  title = {Soliton Distribution in the Ball and Box Cellular Automation Model},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {4},
  number = {1},
  pages = {27-30},
  doi = {10.11648/j.ajmcm.20190401.14},
  url = {https://doi.org/10.11648/j.ajmcm.20190401.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20190401.14},
  abstract = {Solitons are very important waves in Mathematical Physics. They help model many physical situations for instance tsunami waves, application in optical communication systems, application in plasma physics and application in laser guided technology among others. Despite enormous applications, solving the non-linear equations which have soliton solutions is challenging be- cause the nonlinear characteristic of the system abruptly changes due to some slight changes of valid parameters including time. A remedy to this is to use cellular automation models to model soliton dynamics. Cellular automation models do not change in response to slight changes on parameters. In this paper, using computer simulation, an investigation of the distribution of the number of solitons from the Ball and Box cellular automation model was examined. The distribution of the number of solitons from the two and three colour Ball and Box cellular automation was established. Using the online integer sequence, for the two colour Ball and Box cellular automation model, it was found that the distribution of the number of the solitons is indexed by the binomial coefficients. On the Other hand, for the three colour Ball and Box cellular automation model, it was found that only solitons of lengths one to four possesses the distribution while the other soliton lengths do not possess any distribution function.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - Soliton Distribution in the Ball and Box Cellular Automation Model
AU  - Alpha Soko
AU  - James Makungu
Y1  - 2019/06/13
PY  - 2019
N1  - https://doi.org/10.11648/j.ajmcm.20190401.14
DO  - 10.11648/j.ajmcm.20190401.14
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 27
EP  - 30
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20190401.14
AB  - Solitons are very important waves in Mathematical Physics. They help model many physical situations for instance tsunami waves, application in optical communication systems, application in plasma physics and application in laser guided technology among others. Despite enormous applications, solving the non-linear equations which have soliton solutions is challenging be- cause the nonlinear characteristic of the system abruptly changes due to some slight changes of valid parameters including time. A remedy to this is to use cellular automation models to model soliton dynamics. Cellular automation models do not change in response to slight changes on parameters. In this paper, using computer simulation, an investigation of the distribution of the number of solitons from the Ball and Box cellular automation model was examined. The distribution of the number of solitons from the two and three colour Ball and Box cellular automation was established. Using the online integer sequence, for the two colour Ball and Box cellular automation model, it was found that the distribution of the number of the solitons is indexed by the binomial coefficients. On the Other hand, for the three colour Ball and Box cellular automation model, it was found that only solitons of lengths one to four possesses the distribution while the other soliton lengths do not possess any distribution function.
VL  - 4
IS  - 1
ER  -

Copy | Download