Ultradiscrete sine-Gordon Equation over Symmetrized Max-Plus Algebra, and Noncommutative Discrete and Ultradiscrete sine-Gordon Equations
Ultradiscretization with negative values is a long-standing problem and several attempts have been made to solve it. Among others, we focus on the symmetrized max-plus algebra, with which we ultradiscretize the discrete sine-Gordon equation. Another ultradiscretization of the discrete sine-Gordon eq...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2013 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2013
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149360 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Ultradiscrete sine-Gordon Equation over Symmetrized Max-Plus Algebra, and Noncommutative Discrete and Ultradiscrete sine-Gordon Equations / K. Kondo // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 23 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862573170010095616 |
|---|---|
| author | Kondo, K. |
| author_facet | Kondo, K. |
| citation_txt | Ultradiscrete sine-Gordon Equation over Symmetrized Max-Plus Algebra, and Noncommutative Discrete and Ultradiscrete sine-Gordon Equations / K. Kondo // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 23 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Ultradiscretization with negative values is a long-standing problem and several attempts have been made to solve it. Among others, we focus on the symmetrized max-plus algebra, with which we ultradiscretize the discrete sine-Gordon equation. Another ultradiscretization of the discrete sine-Gordon equation has already been proposed by previous studies, but the equation and the solutions obtained here are considered to directly correspond to the discrete counterpart. We also propose a noncommutative discrete analogue of the sine-Gordon equation, reveal its relations to other integrable systems including the noncommutative discrete KP equation, and construct multisoliton solutions by a repeated application of Darboux transformations. Moreover, we derive a noncommutative ultradiscrete analogue of the sine-Gordon equation and its 1-soliton and 2-soliton solutions, using the symmetrized max-plus algebra. As a result, we have a complete set of commutative and noncommutative versions of continuous, discrete, and ultradiscrete sine-Gordon equations.
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| first_indexed | 2025-11-26T06:40:17Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-149360 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-26T06:40:17Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kondo, K. 2019-02-21T07:13:39Z 2019-02-21T07:13:39Z 2013 Ultradiscrete sine-Gordon Equation over Symmetrized Max-Plus Algebra, and Noncommutative Discrete and Ultradiscrete sine-Gordon Equations / K. Kondo // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 23 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37K10; 39A12 DOI: http://dx.doi.org/10.3842/SIGMA.2013.068 https://nasplib.isofts.kiev.ua/handle/123456789/149360 Ultradiscretization with negative values is a long-standing problem and several attempts have been made to solve it. Among others, we focus on the symmetrized max-plus algebra, with which we ultradiscretize the discrete sine-Gordon equation. Another ultradiscretization of the discrete sine-Gordon equation has already been proposed by previous studies, but the equation and the solutions obtained here are considered to directly correspond to the discrete counterpart. We also propose a noncommutative discrete analogue of the sine-Gordon equation, reveal its relations to other integrable systems including the noncommutative discrete KP equation, and construct multisoliton solutions by a repeated application of Darboux transformations. Moreover, we derive a noncommutative ultradiscrete analogue of the sine-Gordon equation and its 1-soliton and 2-soliton solutions, using the symmetrized max-plus algebra. As a result, we have a complete set of commutative and noncommutative versions of continuous, discrete, and ultradiscrete sine-Gordon equations. The author would like to express his gratitude to Professor Tetsuji Tokihiro, who provided
 precise advices with a fine prospect. The author is also grateful to Professor Ralph Willox,
 who provided helpful comments for refining the results. In addition, the author thanks the
 anonymous referees for carefully reading the paper and giving many suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Ultradiscrete sine-Gordon Equation over Symmetrized Max-Plus Algebra, and Noncommutative Discrete and Ultradiscrete sine-Gordon Equations Article published earlier |
| spellingShingle | Ultradiscrete sine-Gordon Equation over Symmetrized Max-Plus Algebra, and Noncommutative Discrete and Ultradiscrete sine-Gordon Equations Kondo, K. |
| title | Ultradiscrete sine-Gordon Equation over Symmetrized Max-Plus Algebra, and Noncommutative Discrete and Ultradiscrete sine-Gordon Equations |
| title_full | Ultradiscrete sine-Gordon Equation over Symmetrized Max-Plus Algebra, and Noncommutative Discrete and Ultradiscrete sine-Gordon Equations |
| title_fullStr | Ultradiscrete sine-Gordon Equation over Symmetrized Max-Plus Algebra, and Noncommutative Discrete and Ultradiscrete sine-Gordon Equations |
| title_full_unstemmed | Ultradiscrete sine-Gordon Equation over Symmetrized Max-Plus Algebra, and Noncommutative Discrete and Ultradiscrete sine-Gordon Equations |
| title_short | Ultradiscrete sine-Gordon Equation over Symmetrized Max-Plus Algebra, and Noncommutative Discrete and Ultradiscrete sine-Gordon Equations |
| title_sort | ultradiscrete sine-gordon equation over symmetrized max-plus algebra, and noncommutative discrete and ultradiscrete sine-gordon equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149360 |
| work_keys_str_mv | AT kondok ultradiscretesinegordonequationoversymmetrizedmaxplusalgebraandnoncommutativediscreteandultradiscretesinegordonequations |