Abstract:
The Neumann and Chaplygin systems on the sphere are simultaneously separable in variables obtained from the standard elliptic coordinates by the proper Bäcklund transformation. We also prove that after similar Bäcklund transformations other curvilinear coordinates on the sphere and on the plane become variables of separation for the system with quartic potential, for the Hénon-Heiles system and for the Kowalevski top. This allows us to speak about some analog of the hetero Bäcklund transformations relating different Hamilton–Jacobi equations.
Keywords:
bi-Hamiltonian geometry, Bäcklund transformations, separation of variables.
\Bibitem{Tsi15}
\by Andrey V. Tsiganov
\paper Simultaneous Separation for the Neumann and Chaplygin Systems
\jour Regul. Chaotic Dyn.
\yr 2015
\vol 20
\issue 1
\pages 74--93
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\crossref{https://doi.org/10.1134/S1560354715010062}
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https://www.mathnet.ru/eng/rcd62
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A V Tsiganov, “Reducible Abelian varieties and Lax matrices for Euler's problem of two fixed centres”, Nonlinearity, 35:10 (2022), 5357
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Gao X.-Y., Guo Y.-J., Shan W.-R., “Scaling Transformations, Hetero-Backlund Transformations and Similarity Reductions on a (2+1)-Dimensional Generalized Variable-Coefficient Boiti-Leon-Pempinelli System For Water Waves”, Rom. Rep. Phys., 73:2 (2021), 111
Zhou T.-Yu., Tian B., Chen S.-S., Wei Ch.-Ch., Chen Yu.-Q., “Backlund Transformations, Lax Pair and Solutions of a Sharma-Tasso-Olver-Burgers Equation For the Nonlinear Dispersive Waves”, Mod. Phys. Lett. B, 35:35 (2021), 2150421
X.-Y. Gao, Y.-J. Guo, W.-R. Shan, “Scaling and hetero-/auto-Backlund transformations with solitons of an extended coupled (2+1)-dimensional Burgers system for the wave processes in hydrodynamics and acoustics”, Mod. Phys. Lett. B, 34:34 (2020), 2050389
X.-Y. Gao, Y.-J. Guo, W.-R. Shan, “Hetero-Backlund transformation and similarity reduction of an extended (2+1)-dimensional coupled Burgers system in fluid mechanics”, Phys. Lett. A, 384:31 (2020), 126788
A. V. Tsiganov, “Reduction of divisors for classical superintegrable gl(3) magnetic chain”, J. Math. Phys., 61:11 (2020), 112703
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A. V. Tsiganov, “Backlund transformations and divisor doubling”, J. Geom. Phys., 126:SI (2018), 148–158
A. V. Tsiganov, “Duffing Oscillator and Elliptic Curve Cryptography”, Nelin. Dinam., 14:2 (2018), 235–241
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Andrey V. Tsiganov, “On Discretization of the Euler Top”, Regul. Chaotic Dyn., 23:6 (2018), 785–796
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A. V. Tsiganov, Springer Proceedings in Mathematics & Statistics, 273, Recent Developments in Integrable Systems and Related Topics of Mathematical Physics, 2018, 47
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