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Lax Pairs for the Discrete Reduced Nahm Systems

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Abstract

We discretise the Lax pair for the reduced Nahm systems and prove its equivalence with the Kahan–Hirota–Kimura discretisation procedure. We show that these Lax pairs guarantee the integrability of the discrete reduced Nahm systems providing an invariant. Also, we show that Nahm systems cannot solve the general problem of characterisation of the integrability for Kahan–Hirota–Kimura discretisations.

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References

  1. Nahm, W.: The construction of all self-dual monopoles by the adhm method monopoles in quantum field theory. In: Craigie, N.S. , Goddard, P., Nahm, W. (eds.) Proc. Monopole Meeting, Trieste 1981, pp. 87 (10pp), Singapore, World Scientific (1982)

  2. Hitchin, N.J., Manton, N.S., Murray, M.K.: Symmetric monopoles. Nonlinearity 8, 661–692 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  3. Ablowitz, M.J., Clarkson, P.A.: Solitons. Nonlinear Evolution Equations and Inverse Scattering. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1991)

    Book  Google Scholar 

  4. Kahan, W.: Unconventional numerical methods for trajectory calculations. Unpublished lecture notes (1993)

  5. Kimura, K., Hirota, R.: Discretization of the Lagrange top. J. Phys. Soc Japan 69, 3193–3199 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  6. Krantz, S.G., Parks, H.R.: Geometric integration theory. Cornerstones. Birkhäuser, Boston, MA (2008)

  7. Petrera, M., Pfadler, A., Suris, Y.U.B.: On integrability of Hirota–Kimura type discretizations: experimental study of the discrete Clebsch system. Exp. Math. 18, 223–247 (2009)

    Article  MathSciNet  Google Scholar 

  8. Petrera, M., Suris, Y.U.B.: On the Hamiltonian structure of Hirota–Kimura discretization of the Euler top. Math. Nachr. 283(11), 1654–1663 (2010)

    Article  MathSciNet  Google Scholar 

  9. Petrera, M., Pfadler, A., Suris, Y.U.B.: On integrability of Hirota–Kimura type discretizations. Regul. Chaot. Dyn. 16, 245–289 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  10. Celledoni, E., McLachlan, R.I., Owren, B., Quispel, G.R.W.: Geometric properties of Kahan’s method, vol. 46 (2013)

  11. Celledoni, E., McLachlan, R.I., McLaren, D.I., Owren, B., Quispel, G.R.W.: Integrability properties of Kahan’s method. J. Phys. A: Math. Theor. 47(36), 365202 (2014)

    Article  MathSciNet  Google Scholar 

  12. Kimura, K.: Lax pair of discrete Nahm equations and its application. In: Proceedings of the International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA), pp. 309–313 (2017)

  13. Sogo, K.: A Lax–Moser pair of Euler’s top. J. Phys. Soc. Jap. 86 (9), 095002 (2017)

    Article  ADS  Google Scholar 

  14. Kimura, K.: A Lax pair of the discrete Euler top. J. Phys. A 50 (24), 245203 (2017). (5pp)

    Article  ADS  MathSciNet  Google Scholar 

  15. Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 21(5), 467–490 (1968)

    Article  MathSciNet  Google Scholar 

  16. Suris, Y.U.B.: The problem of integrable discretization: Hamiltonian approach. Birkhäuser, Basel (2003)

  17. Yu, B.S.: Generalized Toda chains in discrete time. Algebra i Analiz 2(2), 141–157 (1990). In russian

    MathSciNet  MATH  Google Scholar 

  18. Celledoni, E., McLachlan, R.I., McLaren, D.I., Owren, B., Quispel, G.R.W.: Two classes of quadratic vector fields for which the Kahan discretization is integrable. MI Lecture Notes 74, 60–62 (2017)

    Google Scholar 

  19. Petrera, M., Zander, R.: New classes of quadratic vector fields admitting integral-preserving Kahan-Hirota-Kimura discretizations. J. Phys. A Math. Theor. 50(13pp), 205203 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  20. Carstea, A.S., Takenawa, T.: A note on minimization of rational surfaces obtained from birational dynamical systems. J Nonlinear Math. Phys. 20(suppl. 1), 17–33 (2013)

    MathSciNet  MATH  Google Scholar 

  21. van der Kamp, P.H., Celledoni, E., McLachlan, R.I., McLaren, D.I., Owren, B., Quispel, G.R.W.: Three classes of quadratic vector fields for which the Kahan discretisation is the root of a generalised Manin transformation. J. Phys. A Math. Theor. 52(10pp), 045204 (2019)

    Article  ADS  Google Scholar 

  22. Gubbiotti, G., Joshi, N.: Resolution of singularities for a map admitting a bi-quartic invariant. Accepted for publication on Bull. Aust. Math. Soc., preprint on arXiv:2002.10089 (2020)

  23. Calogero, F., Nucci, M.C.: Lax pairs galore. J. Math. Phys. 32(1), 72–74 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  24. Hay, M., Butler, S.: Simple identification of fake Lax pairs. arXiv:1311.2406v1 (2012)

  25. Hay, M., Butler, S.: Two definitions of fake Lax pairs. AIP Conf. Proc. 1648, 180006 (2015)

    Article  Google Scholar 

  26. Golse, F., Mahalov, A., Nicolaenko, B.: Bursting dynamics of the 3D Euler equations in cylindrical domains. In: Instability in Models Connected with Fluid Flows. I, vol. 6 of Int. Math. Ser. (N. Y.), pp. 301–338. Springer, New York (2008)

  27. Falqui, G., Viallet, C.-M.: Singularity, complexity, and quasi-integrability of rational mappings. Comm. Math. Phys. 154, 111–125 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  28. Gubbiotti, G.: A novel integrable fourth-order difference equation admitting three invariants. In: Accepted for Publication in “Proceedings of the Quantum Theory and Symmetries 11” Conference Published in CRM Series on Mathematical Physics, by Springer (2020)

  29. Kimura, K.: From the discrete Nahm equation to the discrete coupled Euler top of KHK version. Private Communication (2021)

  30. Ivanov, R.: Hamiltonian formulation and integrability of a complex symmetric nonlinear system. Phys. Lett. A 350, 232–235 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  31. Celledoni, E., McLachlan, R.I., Owren, B., Quispel, G.R.W.: Discretization of polynomialvector fields by polarisation. Proc. Roy. Soc A 471(10pp.), 20150390 (2015)

    Article  ADS  Google Scholar 

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Acknowledgements

The author expresses his gratitude to Prof. N. Joshi, Prof. G. R. W. Quispel and Dr. D. T. Tran for their helpful discussions during the preparation of this paper. Moreover, we would like to thank Prof. K. Kimura for sharing his results and comments on the coupled Euler system and on the \(\mathfrak {so}\left (4 \right )\) system. Finally, we thank the anonymous referee, whose comments led to a great improvement of the paper.

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  1. School of Mathematics and Statistics F07, The University of Sydney, Sydney, NSW 2006, Australia

    Giorgio Gubbiotti

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  1. Giorgio Gubbiotti
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Correspondence to Giorgio Gubbiotti.

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This research was supported by Dr. M. Radnovič’s grant DP160101728 and by Prof. N. Joshi and Dr. Milena Radnovič’s grant DP200100210 from the Australian Research Council.

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Gubbiotti, G. Lax Pairs for the Discrete Reduced Nahm Systems. Math Phys Anal Geom 24, 9 (2021). https://doi.org/10.1007/s11040-021-09381-7

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