December  2019, 6(2): 277-306. doi: 10.3934/jcd.2019014

Integrable reductions of the dressing chain

1. 

Department of Mathematics, Faculty of Science, University of Hradec Kralove, Czech Republic

2. 

Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus

3. 

Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR 7348 du CNRS, Bât. H3, Boulevard Marie et Pierre Curie, Site du Futuroscope, TSA 61125, 86073 POITIERS Cedex 9, France

* Corresponding author: Charalampos Evripidou

Received  March 2019 Revised  July 2019 Published  November 2019

Fund Project: The research of the first author was supported by the project "International mobilities for research activities of the University of Hradec Králové", CZ.02.2.69/0.0/0.0/16_027/0008487

In this paper we construct a family of integrable reductions of the dressing chain, described in its Lotka-Volterra form. For each $ k, n\in \mathbb N $ with $ n \geqslant 2k+1 $ we obtain a Lotka-Volterra system $ \hbox{LV}_b(n, k) $ on $ \mathbb {R}^n $ which is a deformation of the Lotka-Volterra system $ \hbox{LV}(n, k) $, which is itself an integrable reduction of the $ 2m+1 $-dimensional Bogoyavlenskij-Itoh system $ \hbox{LV}({2m+1}, m) $, where $ m = n-k-1 $. We prove that $ \hbox{LV}_b(n, k) $ is both Liouville and non-commutative integrable, with rational first integrals which are deformations of the rational first integrals of $ \hbox{LV}({n}, {k}) $. We also construct a family of discretizations of $ \hbox{LV}_b(n, 0) $, including its Kahan discretization, and we show that these discretizations are also Liouville and superintegrable.

Citation: Charalampos Evripidou, Pavlos Kassotakis, Pol Vanhaecke. Integrable reductions of the dressing chain. Journal of Computational Dynamics, 2019, 6 (2) : 277-306. doi: 10.3934/jcd.2019014
References:
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M. Adler, P. van Moerbeke and P. Vanhaecke, Algebraic Integrability, Painlevé Geometry and Lie Algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, 47, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-05650-9.  Google Scholar

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P. A. Damianou, C. A. Evripidou, P. Kassotakis and P. Vanhaecke, Integrable reductions of the Bogoyavlenskij-Itoh Lotka-Volterra systems, J. Math. Phys., 58 (2017), 17pp. doi: 10.1063/1.4978854.  Google Scholar

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C. A. EvripidouP. Kassotakis and P. Vanhaecke, Integrable deformations of the Bogoyavlenskij-Itoh Lotka-Volterra systems, Regul. Chaotic Dyn., 22 (2017), 721-739.  doi: 10.1134/S1560354717060090.  Google Scholar

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C. A. EvripidouP. H. van der Kamp and C. Zhang, Dressing the dressing chain, SIGMA Symmetry Integrability Geom. Methods Appl., 14 (2018), 59-73.  doi: 10.3842/SIGMA.2018.059.  Google Scholar

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A. Fordy and A. Hone, Discrete integrable systems and Poisson algebras from cluster maps, Comm. Math. Phys., 325 (2014), 527-584.  doi: 10.1007/s00220-013-1867-y.  Google Scholar

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R. Hirota and K. Kimura, Discretization of the Euler top, J. Phys. Soc. Japan, 69 (2000), 627-630.  doi: 10.1143/JPSJ.69.627.  Google Scholar

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Y. Itoh, A combinatorial method for the vanishing of the Poisson brackets of an integrable Lotka-Volterra system, J. Phys. A, 42 (2009), 11pp. doi: 10.1088/1751-8113/42/2/025201.  Google Scholar

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M. Noumi and Y. Yamada, Affine Weyl groups, discrete dynamical systems and Painlevé equations, Comm. Math. Phys., 199 (1998), 281-295.  doi: 10.1007/s002200050502.  Google Scholar

[19]

P. H. van der Kamp, T. E. Kouloukas, G. R. W. Quispel, D. T. Tran and P. Vanhaecke, Integrable and superintegrable systems associated with multi-sums of products, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 23pp. doi: 10.1098/rspa.2014.0481.  Google Scholar

[20]

A. P. Veselov, Integrable mappings, Russian Math. Surveys, 46 (1991), 1-51.  doi: 10.1070/RM1991v046n05ABEH002856.  Google Scholar

[21]

A. P. Veselov and A. B. Shabat, Dressing chains and the spectral theory of the Schrödinger operator, Funktsional. Anal. i Prilozhen., 27 (1993), 1–21, 96. doi: 10.1007/BF01085979.  Google Scholar

[22]

V. Volterra, Leçons Sur la Théorie Mathématique de la Lutte Pour la Vie, Les Grands Classiques Gauthier-Villars, Éditions Jacques Gabay, Sceaux, 1990.  Google Scholar

show all references

References:
[1]

M. Adler, P. van Moerbeke and P. Vanhaecke, Algebraic Integrability, Painlevé Geometry and Lie Algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, 47, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-05650-9.  Google Scholar

[2]

V. Adler, Cutting of polygons, Funct. Anal. Appl., 27 (1993), 141-143.  doi: 10.1007/BF01085984.  Google Scholar

[3]

O. I. Bogoyavlenskij, Some constructions of integrable dynamical systems, Math. USSR-Izv., 31 (1988), 47-75.  doi: 10.1070/IM1988v031n01ABEH001043.  Google Scholar

[4]

O. I. Bogoyavlenskij, Integrable Lotka-Volterra systems, Regul. Chaotic Dyn., 13 (2008), 543-556.  doi: 10.1134/S1560354708060051.  Google Scholar

[5]

E. Celledoni, R. I. McLachlan, D. I. McLaren, B. Owren and G. R. W. Quispel, Integrability properties of Kahan's method, J. Phys. A, 47 (2014), 20pp. doi: 10.1088/1751-8113/47/36/365202.  Google Scholar

[6]

P. A. Damianou, C. A. Evripidou, P. Kassotakis and P. Vanhaecke, Integrable reductions of the Bogoyavlenskij-Itoh Lotka-Volterra systems, J. Math. Phys., 58 (2017), 17pp. doi: 10.1063/1.4978854.  Google Scholar

[7]

C. A. EvripidouP. Kassotakis and P. Vanhaecke, Integrable deformations of the Bogoyavlenskij-Itoh Lotka-Volterra systems, Regul. Chaotic Dyn., 22 (2017), 721-739.  doi: 10.1134/S1560354717060090.  Google Scholar

[8]

C. A. EvripidouP. H. van der Kamp and C. Zhang, Dressing the dressing chain, SIGMA Symmetry Integrability Geom. Methods Appl., 14 (2018), 59-73.  doi: 10.3842/SIGMA.2018.059.  Google Scholar

[9]

A. Fordy and A. Hone, Discrete integrable systems and Poisson algebras from cluster maps, Comm. Math. Phys., 325 (2014), 527-584.  doi: 10.1007/s00220-013-1867-y.  Google Scholar

[10]

R. Hirota and K. Kimura, Discretization of the Euler top, J. Phys. Soc. Japan, 69 (2000), 627-630.  doi: 10.1143/JPSJ.69.627.  Google Scholar

[11]

R. Hirota and K. Kimura, Discretization of the Lagrange top, J. Phys. Soc. Japan, 69 (2000), 3193-3199.  doi: 10.1143/JPSJ.69.3193.  Google Scholar

[12]

Y. Itoh, Integrals of a Lotka-Volterra system of odd number of variables, Progr. Theoret. Phys., 78 (1987), 507-510.  doi: 10.1143/PTP.78.507.  Google Scholar

[13]

Y. Itoh, A combinatorial method for the vanishing of the Poisson brackets of an integrable Lotka-Volterra system, J. Phys. A, 42 (2009), 11pp. doi: 10.1088/1751-8113/42/2/025201.  Google Scholar

[14]

T. E. Kouloukas, G. R. W. Quispel and P. Vanhaecke, Liouville integrability and superintegrability of a generalized Lotka-Volterra system and its Kahan discretization, J. Phys. A, 49 (2016), 13pp. doi: 10.1088/1751-8113/49/22/225201.  Google Scholar

[15]

C. Laurent-Gengoux, A. Pichereau and P. Vanhaecke, Poisson Structures, Fundamental Principles of Mathematical Sciences, 347, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-31090-4.  Google Scholar

[16]

A. J. Lotka, Analytical Theory of Biological Populations, The Plenum Series on Demographic Methods and Population Analysis, Plenum Press, New York, 1998. doi: 10.1007/978-1-4757-9176-1.  Google Scholar

[17]

A. S. Miscenko and A. T. Fomenko, Generalized Liouville method for the integration of Hamiltonian systems, Funkcional. Anal. i Priložen., 12 (1978), 46–56, 96. doi: 10.1007/BF01076254.  Google Scholar

[18]

M. Noumi and Y. Yamada, Affine Weyl groups, discrete dynamical systems and Painlevé equations, Comm. Math. Phys., 199 (1998), 281-295.  doi: 10.1007/s002200050502.  Google Scholar

[19]

P. H. van der Kamp, T. E. Kouloukas, G. R. W. Quispel, D. T. Tran and P. Vanhaecke, Integrable and superintegrable systems associated with multi-sums of products, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 23pp. doi: 10.1098/rspa.2014.0481.  Google Scholar

[20]

A. P. Veselov, Integrable mappings, Russian Math. Surveys, 46 (1991), 1-51.  doi: 10.1070/RM1991v046n05ABEH002856.  Google Scholar

[21]

A. P. Veselov and A. B. Shabat, Dressing chains and the spectral theory of the Schrödinger operator, Funktsional. Anal. i Prilozhen., 27 (1993), 1–21, 96. doi: 10.1007/BF01085979.  Google Scholar

[22]

V. Volterra, Leçons Sur la Théorie Mathématique de la Lutte Pour la Vie, Les Grands Classiques Gauthier-Villars, Éditions Jacques Gabay, Sceaux, 1990.  Google Scholar

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