How One can Repair Non-integrable Kahan Discretizations. II. A Planar System with Invariant Curves of Degree 6
Tóm tắt
We find a novel one-parameter family of integrable quadratic Cremona maps of the plane preserving a pencil of curves of degree 6 and of genus 1. They turn out to serve as Kahan-type discretizations of a novel family of quadratic vector fields possessing a polynomial integral of degree 6 whose level curves are of genus 1, as well. These vector fields are non-homogeneous generalizations of reduced Nahm systems for magnetic monopoles with icosahedral symmetry, introduced by Hitchin, Manton and Murray. The straightforward Kahan discretization of these novel non-homogeneous systems is non-integrable. However, this drawback is repaired by introducing adjustments of order
$$O(\epsilon ^2)$$
in the coefficients of the discretization, where
$$\epsilon $$
is the stepsize.
Tài liệu tham khảo
Bellon, M.P., Viallet, C.-M.: Algebraic entropy. Commun. Math. Phys. 204(2), 425–437 (1999)
Celledoni, E., McLachlan, R.I., Owren, B., Quispel, G.R.W.: Geometric properties of Kahan’s method. J. Phys. A 46, 025201 (2013)
Celledoni, E., McLachlan, R.I., McLaren, D.I., Owren, B., Quispel, G.R.W.: Integrability properties of Kahan’s method. J. Phys. A 47, 365202 (2014)
Celledoni, E., McLachlan, R.I., McLaren, D.I., Owren, B., Quispel, G.R.W.: Discretization of polynomial vector fields by polarization. Proc. R. Soc. A 471, 20150390 (2015). (10 pp)
Celledoni, E., McLachlan, R.I., McLaren, D.I., Owren, B., Quispel, G.R.W.: Two classes of quadratic vector fields for which the Kahan map is integrable. MI Lecture Note, Kyushu University, vol. 74, pp. 60–62 (2016)
Celledoni, E., McLaren, D.I., Owren, B., Quispel, G.R.W.: Geometric and integrability properties of Kahan’s method: the preservation of certain quadratic integrals. J. Phys. A 52, 065201 (2019)
Diller, J.: Dynamics of birational maps of \({\mathbb{P}}^2\). Indiana Univ. Math. J. 45(3), 721–772 (1996)
Diller, J.: Cremona transformations, surface automorphisms, and plane cubics. Michigan Math. J. 60, 409–440 (2011)
Diller, J., Favre, C.: Dynamics of bimeromorphic maps of surfaces. Am. J. Math. 123(6), 1135–1169 (2001)
Duistermaat, J.J.: Discrete Integrable Systems. QRT Maps and Elliptic Surfaces. Springer, New York (2010)
Hirota, R.: Nonlinear partial difference equations. I. A difference analogue of the Korteweg-de Vries equation. J. Phys. Soc. Jpn. 43, 1423–1433 (1977)
Hirota, R.: Nonlinear partial difference equations. II. Discrete time Toda equation. J. Phys. Soc. Jpn. 43, 2074–2078 (1977)
Hirota, R.: Nonlinear partial difference equations. III. Discrete sine-Gordon equation. J. Phys. Soc. Jpn. 43, 2079–2086 (1977)
Hirota, R.: Nonlinear partial difference equations. IV. Bäcklund transformation for the discrete-time Toda equation. J. Phys. Soc. Jpn. 45, 321–332 (1978)
Hirota, R.: Nonlinear partial difference equations. V. Nonlinear equations reducible to linear equations. J. Phys. Soc. Jpn. 46, 312–319 (1979)
Hirota, R.: Discrete analogue of a generalized Toda equation. J. Phys. Soc. Jpn. 50, 3785–3791 (1981)
Hirota, R., Kimura, K.: Discretization of the Euler top. J. Phys. Soc. Jpn. 69(3), 627–630 (2000)
Hitchin, N.J., Manton, N.S., Murray, M.K.: Symmetric monopoles. Nonlinearity 8(5), 661–692 (1995)
Hone, A.N.W., Quispel, G.R.W.: Analogues of Kahan’s method for higher order equations of higher degree. In: Nijhoff, F., Shi, Y., Zhang, D. (eds.) Asymptotic, Algebraic and geometric Aspects of Integrable Systems, pp. 175–189. Springer, New York (2020)
Kahan, W.: Unconventional numerical methods for trajectory calculations. Unpublished Lecture Notes (1993)
Kimura, K., Hirota, R.: Discretization of the Lagrange top. J. Phys. Soc. Jpn. 69(10), 3193–3199 (2000)
Mickens, R.E.: Nonstandard Finite Difference Models of Differential Equations. World Scientific, Singapore (1994)
Petrera, M., Pfadler, A., Suris, Yu.B.: On integrability of Hirota–Kimura-type discretizations: experimental study of the discrete Clebsch system. Exp. Math. 18(2), 223–247 (2009)
Petrera, M., Pfadler, A., Suris, Y.B.: On integrability of Hirota–Kimura type discretizations. Regular Chaotic Dyn. 16(3–4), 245–289 (2011)
Petrera, M., Pfadler, A., Suris, Y.B.: On the construction of elliptic solutions of integrable birational maps. Exp. Math. 26(3), 324–341 (2017). ((with appendix by Y.N. Fedorov))
Petrera, M., Suris, Y.B.: On the Hamiltonian structure of Hirota–Kimura discretization of the Euler top. Math. Nachr. 283(11), 1654–1663 (2010)
Petrera, M., Suris, Yu.B.: A construction of a large family of commuting pairs of integrable symplectic birational 4-dimensional maps. Proc. R. Soc. A 473, 20160535 (2017). (16 pp)
Petrera, M., Suris, Y.B.: New results on integrability of the Kahan–Hirota–Kimura discretizations. In: Euler, N. (ed.) Nonlinear Systems and Their Remarkable Mathematical Structures, pp. 94–120. CRC Press, Boca Raton, FL (2018)
Petrera, M., Suris, Y.B.: Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. II. Systems with a linear Poisson tensor. J. Comput. Dyn. 6, 401–408 (2019)
Petrera, M., Smirin, J., Suris, Yu.B.: Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Proc. R. Soc. A 475, 20180761 (2019). (13 pp)
Petrera, M., Suris, Y.B., Wei, K., Zander, R.: Manin involutions for elliptic pencils and discrete integrable systems. Math. Phys. Anal. Geom. 24, 6 (2021)
Petrera, M., Zander, R.: New classes of quadratic vector fields admitting integral-preserving Kahan–Hirota–Kimura discretizations. J. Phys. A Math. Theor. 50, 205203 (2017). (13 pp)
Petrera, M., Suris, Y.B., Zander, R.: How one can repair non-integrable Kahan discretizations. J. Phys. A Math. Theor. 53, 37LT01 (2020)
Quispel, G.R.W., McLaren, D.I., van der Kamp, P.H.: A novel 8-parameter integrable map in \({\mathbb{R}}^4\). J. Phys. A Math. Theor. 53, 40LT01 (2020)
Quispel, G.R.W., Roberts, J.A.G., Thompson, C.J.: Integrable mappings and soliton equations II. Physica D 34, 183–192 (1989)
Sanz-Serna, J.M.: An unconventional symplectic integrator of W. Kahan. Appl. Numer. Math. 16, 245–250 (1994)
Suris, Y.B.: The Problem of Integrable Discretization: Hamiltonian Approach. Progress in Mathematics, vol. 219. Birkhäuser, Basel (2003)
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 discretization is the root of a generalised Manin transformation. J. Phys. A Math. Theor. 52, 045204 (2019)
van der Kamp, P.H., McLaren, D.I., Quispel, G.R.W.: Generalised Manin transformations and QRT maps. J. Comput. Dyn. 8(2), 183–211 (2021)
Zander, R.: On the singularity structure of Kahan discretizations of a class of quadratic vector fields. Eur. J. Math. 7, 1046–1073 (2021)