2007 Fiscal Year Final Research Report Summary
q-analogue of the ADHMN construction and the self-dual Yang-Mills fields
| Project/Area Number |
16540352
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Mathematical physics/Fundamental condensed matter physics
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| Research Institution | Kisarazu National College of Technology |
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Principal Investigator |
KAMATA Masaru Kisarazu National College of Technology, Natural Science Education, Professor (10169609)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAMURA Atsushi Kitasato University, Department of Physics, School of Science, Lecturer (90245415)
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| Project Period (FY) |
2004 – 2007
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| Keywords | Yang-Mills fields / Self-dual / ADHMN construction / q-analogue / Nahm equation / Integrable system / Singularity confinement test / Algebraic entropy |
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| Research Abstract |
In mathematical physics, the ADHM (ADHMN) construction is a well-known method to construct most generally the (anti-) selfdual Yang-Mills fields in the four-dimensional Euclidean space R^4 (R×R^3). The head investigator and the investigator presented, in a previous paper, a q-analogue of the ADHMN construction based on the countably infinite dimensional Hilbert space l^2 defined at a discrete q-interval I_q:={±1/2, ±q/2, ±^2/2, ., ±q^n/2, .} with a real parameter q. Further, as a q-analogue of the Nahm equations which describe the multi-monopole configurations in R×R^3, the following q-discretized Nahm equations (or q-Nahm equations in short) are obtained:
D_qT_j(z)=(1/2)ε_{jkl}(T_k(qz)T_l(z)-T_l(qz)T:_k(z)), j,k,l=1,2,3
where D_q is the q-difference operator.
The following have been studied in this project in the above term :
1. The Euler-top type reduction of the q-Nahm equations yields a system of difference equations which pass the singularity confinement test.
2. A Toda-Flaschka type reduction of the q-Nahm equations yields a system of difference equations with at least one conserved quantity.
3. A q-analogue of the caloron solution in S^l×R^3 is obtained through the ADHM construction.
4. Relations between the Nahm equations and the Euler top equations are studied.
5. Singular solutions derived from the known solutions to the CBS equations are investigated.
6. An extended KP hierarchy with the Riemann-Liouville differential-integral operators of the order 1/3 is examined.
7. A preliminary calculation of the algebraic entropy to the q-Nahm equations is given.
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Research Products
(30 results)
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[Presentation] Aspects of q-discretized Nahm equations2007
Author(s)
Atsushi Nakamula
Organizer
The Fifth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena:Computation and Theory, Oral presentation
Place of Presentation
University of Georgia,Athens,GA,USA
Year and Date
2007-04-18
Description
「研究成果報告書概要(和文)」より
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[Presentation] Aspects of q-discretized Nahm equations2007
Author(s)
Atsushi Nakamula
Organizer
The Fifth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena : Computation and Theory, Oral presentation
Place of Presentation
University of Georgia, Athens, USA
Year and Date
2007-04-18
Description
「研究成果報告書概要(欧文)」より
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[Presentation] q-discretization of Nahm equations and its analysis2006
Author(s)
Atsushi Nakamula, Masaru Kamata
Organizer
Proceedings of Symposium on Crossover among Theoretical, Numerical and Experimental Studies on Nonlinear Waves, Poster presentation, Research Institute for Applied Mathematics
Place of Presentation
Kvushvn Uniwrci
Year and Date
2006-11-07
Description
「研究成果報告書概要(欧文)」より
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[Presentation] On q-discretized Nahm equations2006
Author(s)
Atsushi Nakamula
Organizer
The 6th International Conference on Soliton and Integrable Systems, Oral presentation
Place of Presentation
Northwest University, Xian, People's Republic of China
Year and Date
2006-10-05
Description
「研究成果報告書概要(欧文)」より
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[Presentation] Integrability of q-discretized Nahm Equations2006
Author(s)
M.Kamata and A.Nakamula
Organizer
SIDE VII (The 7th International Meeting of Symmetries and Integrability of Difference Equations), Poster presentation
Place of Presentation
Melbourne University,Melbourne,Australia
Year and Date
2006-07-11
Description
「研究成果報告書概要(和文)」より
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[Presentation] Integrability of q-discretized Nahm equations2005
Author(s)
Atsushi Nakamula, Masaru Kamata
Organizer
Symposium on Fhenomena and Mathematics of Nonlinear Waves and Dynamical Systems, Posterpresentation, Research Institute for Applied Mathematics
Place of Presentation
Kyushyu University
Year and Date
2005-11-11
Description
「研究成果報告書概要(欧文)」より
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[Presentation] Integrability of q-discretized Nahm Equations2005
Author(s)
Masaru Kamata, Atsushi Nakamula
Organizer
SIDE VII(The 7' International Meeting of Symmetries and Integrability of Difference Equations), Poster presentation
Place of Presentation
Melbourne University, Melbourne, Australia
Year and Date
2005-07-11
Description
「研究成果報告書概要(欧文)」より
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