zbMATH — the first resource for mathematics

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.
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.)
Full Text: DOI
[1] Dunajski, M.; Kryński, W., Einstein-Weyl geometry dispersionless Hirota equation and Veronese webs, Math. Proc. Cambridge Philos. Soc., 157, 1, 139-150, (2014) · Zbl 1296.53036
[2] Kruglikov, B.; Panasyuk, A., Veronese webs and nonlinear pdes, J. Geom. Phys., 115, 45-60, (2017) · Zbl 1376.53026
[3] Dunajski, M.; Mason, L.; Tod, P., Einstein-Weyl geometry the dkp equation and twistor theory, J. Geom. Phys., 37, 63-93, (2001) · Zbl 0990.53052
[4] Godliński, M.; Nurowski, P., On three-dimensional Weyl structures with reduced holonomy, Classical Quantum Gravity, 23, 603-608, (2006) · Zbl 1087.53043
[5] Manakov, S.; Santini, P., The Cauchy problem on the plane for the dispersionless Kadomtsev-Petviashvili equation, JETP Lett., 83, 462-466, (2006)
[6] Ferapontov, E.; Kruglikov, B., Dispersionless integrable systems in 3D and Einstein-Weyl geometry, J. Differential Geom., 97, 215-254, (2014) · Zbl 1306.37084
[7] Ferapontov, E.; Moss, J., Linearly degenerate PDEs and quadratic line complexes, Comm. Anal. Geom., 23, 1, 91-127, (2015) · Zbl 1310.53013
[8] Dunajski, M., A class of Einstein-Weyl spaces associated to an integrable system of hydrodynamic type, J. Geom. Phys., 51, 126-137, (2004) · Zbl 1110.53032
[9] Dunajski, M.; Mason, L., Hyper-Kähler hierarchies and their twistor theory, Comm. Math. Phys., 213, 641-672, (2000) · Zbl 0988.53021
[10] Calderbank, D., Integrable background geometries, SIGMA, 10, (2014) · Zbl 1288.53074
[11] Bryant, R., Two exotic holonomies in dimension four, path geometries, and twistor theory, Amer. Math. Soc. Proc. Symp. Pure Math., 53, 33-88, (1991) · Zbl 0758.53017
[12] Dunajski, M.; Tod, P., Paraconformal geometry of \(n\)th order ODEs and exotic holonomy in dimension four, J. Geom. Phys., 56, 1790-1809, (2006) · Zbl 1096.53028
[13] E. Ferapontov, B. Kruglikov, Dispersionless integrable hierarchies and \(G L(2, \mathbb{R})\) geometry, (2016). arXiv:1607.01966 [nlin.SI].
[14] Godliński, M.; Nurowski, P., \(G L(2, \mathbb{R})\)-geometry of ODE’s, J. Geom. Phys., 60201, 991-1027, (2010) · Zbl 1260.53030
[15] Kryński, W., Paraconformal structures ordinary differential equations and totally geodesic manifolds, J. Geom. Phys., 103, 1-19, (2016) · Zbl 1432.53028
[16] Nurowski, P., Differential equations and conformal structures, J. Geom. Phys., 55, 1, 19-49, (2004) · Zbl 1082.53024
[17] W. Kryński, T. Mettler, GL(2)-structures in dimension four, H-flatness and integrability, Comm. Anal. Geom. (2016) in press. arXiv:1611.08228.
[18] Martinez Alonso, L.; Shabat, A., Towards a theory of differential constraints of a hydrodynamic hierarchy, J. Nonlinear Math. Phys., 10, 229-242, (2003) · Zbl 1055.35092
[19] Martinez Alonso, L.; Shabat, A., Hydrodynamic reductions and solutions of a universal hierarchy, Teoret. Mat. Fiz., 140, 216-229, (2004) · Zbl 1178.37067
[20] Hitchin, N., Complex manifolds and einstein’s equations, (Twistor Geometry and Non-Linear Systems, Lecture Notes in Mathematics, vol. 970, (1982)) · Zbl 0507.53025
[21] Tod, P., Einstein-Weyl spaces and third-order differential equation, J. Math. Phys., 41, 5572, (2000) · Zbl 0979.53050
[22] Gelfand, I.; Zakharevich, I.; Webs, ., Veronese curves and bi-Hamiltonian systems, J. Funct. Anal., 99, 150-178, (1991)
[23] Gelfand, I.; Zakharevich, I., Webs, lenard schemes and the local geometry of bi-Hamiltonian Toda and Lax structures, Selecta Math., 6, 131-183, (2000) · Zbl 0986.37060
[24] I. Zakharevich, Nonlinear wave equation, nonlinear Riemann problem, and the twistor transform of Veronese webs, (2000). arXiv:math-ph/0006001.
[25] Morozov, O.; Pavlov, M., Bäcklund transformations between four Lax-integrable 3D equations, J. Nonlinear Math. Phys., 24, 4, 465-468, (2017)
[26] Kryński, W., Webs and the plebański equations, Math. Proc. Cambridge Philos. Soc., 161, 3, 455-468, (2016) · Zbl 1371.53015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.