Abstract:
We consider analogues of auto- and hetero-Bäcklund transformations for the Jacobi system on a three-axes ellipsoid. Using the results in a Weierstrass paper, where the change of times reduces integrating the equations of motion to inverting the Abel mapping, we construct the differential Abel equations and auto-Bäcklund transformations preserving the Poisson bracket with respect to which the equations of motion written in the Weierstrass form are Hamiltonian. Transforming this bracket to the canonical form, we can construct a new integrable system on the ellipsoid with a Hamiltonian of the natural form and with a fourth-degree integral of motion in momenta.
Keywords:
integrable system, Bäcklund transformation, Jacobi system on an ellipsoid.
Citation:
A. V. Tsiganov, “Bäcklund transformations for the Jacobi system on an ellipsoid”, TMF, 192:3 (2017), 473–488; Theoret. and Math. Phys., 192:3 (2017), 1350–1364
This publication is cited in the following 16 articles:
Andrey V. Tsiganov, “Integrable Systems on a Sphere, an Ellipsoid and a Hyperboloid”, Regul. Chaotic Dyn., 28:6 (2023), 805–821
Xin-Yi Gao, “Considering the wave processes in oceanography, acoustics and hydrodynamics by means of an extended coupled (2+1)-dimensional Burgers system”, Chinese Journal of Physics, 86 (2023), 572
Xin-Yi Gao, Yong-Jiang Guo, Wen-Rui Shan, “Oceanic shallow-water symbolic computation on a (2+1)-dimensional generalized dispersive long-wave system”, Physics Letters A, 457 (2023), 128552
Andrey V. Tsiganov, “Equivalent Integrable Metrics on the Sphere with Quartic Invariants”, SIGMA, 18 (2022), 094, 19 pp.
X.-Y. Gao, Y.-J. Guo, W.-R. Shan, “Symbolic computation on a (2+1)-dimensional generalized variable-coefficient Boiti-Leon-Pempinelli system for the water waves”, Chaos Solitons Fractals, 150 (2021), 111066
X.-Y. Gao, Y.-J. Guo, W.-R. Shan, “In oceanography, acoustics and hydrodynamics: an extended coupled (2+1) -dimensional Burgers system”, Chin. J. Phys., 70 (2021), 264–270
B. Zhang, E. Fan, “Riemann-Hilbert approach for a Schrödinger-type equation with nonzero boundary conditions”, Mod. Phys. Lett. B, 35:12 (2021), 2150208
X.-Y. Gao, Y.-J. Guo, W.-R. Shan, “Oceanic studies via a variable-coefficient nonlinear dispersive-wave system in the solar system”, Chaos Solitons Fractals, 142 (2021), 110367
X.-Y. Gao, Y.-J. Guo, W.-R. Shan, “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
A. V. Tsiganov, “Discretization and superintegrability all rolled into one”, Nonlinearity, 33:9 (2020), 4924–4939
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
X.-Y. Gao, Y.-J. Guo, W.-R. Shan, “Shallow water in an open sea or a wide channel: auto- and non-auto-Backlund transformations with solitons for a generalized (2+1)-dimensional dispersive long-wave system”, Chaos Solitons Fractals, 138 (2020), 109950
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
A. V. Tsiganov, “Backlund transformations and divisor doubling”, J. Geom. Phys., 126:SI (2018), 148–158
A. V. Tsiganov, “On exact discretization of cubic-quintic duffing oscillator”, J. Math. Phys., 59:7 (2018), 072703
A. V. Tsiganov, “Discretization of Hamiltonian systems and intersection theory”, Theoret. and Math. Phys., 197:3 (2018), 1806–1822
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