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Principal symmetric space analysis
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 |
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.
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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