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

Integrable reductions of the dressing chain

Charalampos Evripidou 1,2,, , Pavlos Kassotakis 2, and  Pol Vanhaecke 3,

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.

Keywords: Integrable systems, deformations, discretizations.
Mathematics Subject Classification: 53D17, 70H06.
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:
[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.

[2]

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

[3]

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

[4]

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

[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.

[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.

[7]

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

[8]

C. A. Evripidou, P. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[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.

[22]

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

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.

[2]

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

[3]

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

[4]

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

[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.

[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.

[7]

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

[8]

C. A. Evripidou, P. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[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.

[22]

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

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