Aspects of \(q\)-discretized Nahm equations. (English) Zbl 1179.39012
Summary: There is a conjecture by R. S. Ward [Philos. Trans. R. Soc. Lond., A 315, 451–457 (1985; Zbl 0579.35078)] that almost all of integrable equations are derived from (anti-)self-dual (ASD) Yang-Mills equations. This conjecture is supported by many concrete examples, e.g., the Nahm equations [W. Nahm, Phys. Lett. B 90, No. 4, 413–414 (1980); in: Monopoles in quantum field theory. Proceedings of the Monopole Meeting, held at Trieste, Italy, 11–15 December 1981. Singapore (Singapore): World Scientific Publishing, 87–94 (1982)].
In this work, we consider a situation that if the ASD conditions are slightly loosened, as to how it affects the integrability of the equations. For this purpose, we consider a \(q\)-analog of the Nahm equations, as a non-ASD system. The analysis is performed on the reduced system which is a \(q\)-analog of the Euler-Arnold top, by the singularity confinement test and the estimation of the algebraic entropy.
In this work, we consider a situation that if the ASD conditions are slightly loosened, as to how it affects the integrability of the equations. For this purpose, we consider a \(q\)-analog of the Nahm equations, as a non-ASD system. The analysis is performed on the reduced system which is a \(q\)-analog of the Euler-Arnold top, by the singularity confinement test and the estimation of the algebraic entropy.
MSC:
| 39A13 | Difference equations, scaling (\(q\)-differences) |
Citations:
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