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On deformations of the dispersionless Hirota equation. (English) Zbl 1392.35263
The class of hyper-CR Einstein-Weyl structures on \(\mathbb{R}^3\) can be described in terms of the solutions to the dispersionless Hirota equation. The aim of this paper is to show that simple geometric constructions on the associated twistor space lead to deformations of the Hirota equation that have been introduced recently by B. Kruglikov and A. Panasyuk [ibid. 115, 45–60 (2017; Zbl 1376.53026)]. The method produces also the hyper-CR equation and can be applied to other geometric structures related to different twistor constructions. The paper explores ideas of M. Dunajski and the author [Math. Proc. Camb. Philos. Soc. 157, No. 1, 139–150 (2014; Zbl 1296.53036), Section 3] and interprets the results of [Kruglikov and Panasyuk, loc. cit.] from the point of view of the geometry of the twistor space associated to the Einstein-Weyl structures. Moreover, the author extends the results obtained in [Kruglikov and Panasyuk, loc. cit.]. The constructions presented in the present paper can be also applied to other dispersionless equations related to different twistorial constructions. The author analyzes two examples at the end of the paper. Namely he considers systems describing the hyper-Hermitian structures and the Veronese webs in dimension 4 (which easily generalize to higher dimensions). It is shown in [the author, J. Geom. Phys. 103, 1–19 (2016; Zbl 1432.53028)] that the Veronese webs are described by a hierarchy of integrable systems that generalizes the Hirota equation (integrable systems describing more general classes of \(GL(2)\)-structures are given in [E. V. Ferapontov and B. Kruglikov, “Dispersionless integrable hierarchies and \(\mathrm{GL}(2,R)\) geometry”, Preprint, arXiv:1607.01966; the author and T. Mettler, “\(\mathrm{GL}(2)\)-structures in dimension four, \(H\)-flatness and integrability”, Preprint, arXiv:1611.08228]. In the present paper, the author proves that the webs can be described by two other hierarchies.
MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
53A60 Differential geometry of webs
53C28 Twistor methods in differential geometry
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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