Our brief communication is devoted to the study of differential-geometric structure and the Lax–Sato integrability of the reduced Shabat-type, Hirota, and Kupershmidt heavenly equations.
References
L. M. Alonso and A. B. Shabat, “Hydrodynamic reductions and solutions of a universal hierarchy,” Theor. Math. Phys., 104, 1073–1085 (2004).
M. Dunajski and W. Kryński, Einstein–Weyl Geometry, Dispersionless Hirota Equation and Veronese Webs, arXiv:1301.0621.
O. E. Hentosh, Ya. A. Prykarpatsky, D. Blackmore, and A. K. Prykarpatski, “Lie-algebraic structure of Lax–Sato integrable heavenly equations and the Lagrange–d’Alembert principle,” J. Geom. Phys., 120, 208–227 (2017).
O. I. Morozov and A. Sergyeyev, The Four-Dimensional Martinez–Alonso–Shabat Equation: Reductions, Nonlocal Symmetries, and a Four-Dimensional Integrable Generalization of the ABC Equation (2014), (Preprint submitted to the JGP).
M. Pavlov, “Kupershmidt hydrodynamic chains and lattices,” Int. Math. Res. Not. IMRN, 1–43 (2006).
Ya. A. Prykarpatskyy and A. M. Samoilenko, “Classical M. A. Buhl problem, its Pfeiffer–Sato solutions, and the classical Lagrange–d’Alembert principle for the integrable heavenly-type nonlinear equations,” Ukr. Mat. Zh., 69, No. 12, 1652–1689 (2017); English translation : Ukr. Math. J., 69, No. 2, 1924–1967 (2017).
B. Szablikowski and M. Błaszak, “Meromorphic Lax representations of (1+1)-dimensional multi-Hamiltonian dispersionless systems,” J. Math. Phys., 47, No. 9 (2006).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, No. 2, pp. 293–296, February, 2018.
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Prytula, M.M., Hentosh, O.E. & Prykarpatskyy, Y.A. Differential-Geometric Structure and the Lax–Sato Integrability of a Class of Dispersionless Heavenly-Type Equations. Ukr Math J 70, 334–339 (2018). https://doi.org/10.1007/s11253-018-1503-2
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DOI: https://doi.org/10.1007/s11253-018-1503-2