Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T03:59:44.603Z Has data issue: false hasContentIssue false
  • English
  • Français

Dispersionless integrable hierarchies and GL(2, ℝ) geometry

Part of: Equations of mathematical physics and other areas of application Local differential geometry Global differential geometry Infinite-dimensional Hamiltonian systems Classical differential geometry

Published online by Cambridge University Press:  08 October 2019

EVGENY FERAPONTOV  and
BORIS KRUGLIKOV
EVGENY FERAPONTOV
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU. Institute of Mathematics, Ufa Federal Research Centre, Russian Academy of Sciences, Ufa 450008, Russia. e-mail: [email protected]
BORIS KRUGLIKOV
Affiliation:
Department of Mathematics and Statistics, UiT the Arctic University of Norway, Tromsø 9037, Norway. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Paraconformal or GL(2, ℝ) geometry on an n-dimensional manifold M is defined by a field of rational normal curves of degree n – 1 in the projectivised cotangent bundle ℙT*M. Such geometry is known to arise on solution spaces of ODEs with vanishing Wünschmann (Doubrov–Wilczynski) invariants. In this paper we discuss yet another natural source of GL(2, ℝ) structures, namely dispersionless integrable hierarchies of PDEs such as the dispersionless Kadomtsev–Petviashvili (dKP) hierarchy. In the latter context, GL(2, ℝ) structures coincide with the characteristic variety (principal symbol) of the hierarchy.

Dispersionless hierarchies provide explicit examples of particularly interesting classes of involutive GL(2, ℝ) structures studied in the literature. Thus, we obtain torsion-free GL(2, ℝ) structures of Bryant [5] that appeared in the context of exotic holonomy in dimension four, as well as totally geodesic GL(2, ℝ) structures of Krynski [33]. The latter possess a compatible affine connection (with torsion) and a two-parameter family of totally geodesic α-manifolds (coming from the dispersionless Lax equations), which makes them a natural generalisation of the Einstein–Weyl geometry.

Our main result states that involutive GL(2, ℝ) structures are governed by a dispersionless integrable system whose general local solution depends on 2n – 4 arbitrary functions of 3 variables. This establishes integrability of the system of Wünschmann conditions.

MSC classification

Primary: 35Q51: Soliton-like equations 37K10: Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.) 37K25: Relations with differential geometry 53A40: Other special differential geometries 53B05: Linear and affine connections 53B50: Applications to physics 53C26: Hyper-Kähler and quaternionic Kähler geometry, “special” geometry 53C80: Applications to physics
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 1 , January 2021 , pp. 129 - 154
Copyright
© Cambridge Philosophical Society 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Adler, V. E. and Shabat, A. B.. A model equation of the theory of solitons. Theoret. and Math. Phys. 153, no. 1 (2007), 1373–1387.CrossRefGoogle Scholar
Akhmedova, V. and Zabrodin, A.. Elliptic parameterisation of Pfaff integrable hierarchies in the zero-dispersion limit. Teoret. Mat. Fiz. 185, no. 3 (2015), 410–422.Google Scholar
Berger, M.. Surles groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes. Bull. Soc. Math. France 83 (1955), 279–330.CrossRefGoogle Scholar
Bogdanov, L. V.. Doubrov-Ferapontov general heavenly equation and the hyper-Kähler hierarchy. J. Phys. A 48, no. 23 (2015), 235202, 15 pp.CrossRefGoogle Scholar
Bryant, R. L.. Two exotic holonomies in dimension four, path geometries, and twistor theory. In: Complex geometry and Lie theory (Sundance, UT, 1989), 33–88, Proc. Sympos. Pure Math., 53. (Amer. Math. Soc., Providence, RI 1991).CrossRefGoogle Scholar
Bryant, R.. Classical, exceptional, and exotic holonomies: a status report. Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), 93–165, Sémin. Congr., 1 (Soc. Math. France, Paris, 1996).Google Scholar
Bryant, R. L., Involutive rational normal structures. www.cim.nankai.edu.cn/activites/conferences/hy20090511/pdf/BryantConfSlides.pdf (2009).Google Scholar
Bryant, R. L., Chern, S. S., Gardner, R. B., Goldschmidt, H. L. and Griffiths, P. A.. Exterior differential systems. MSRI Publications 18 (Springer-Verlag 1991).CrossRefGoogle Scholar
Calderbank, D. M. J., Integrable background geometries. SIGMA 10 (2014), Paper 034, 51 pp; see also http://people.bath.ac.uk/dmjc20/mpapers.html (2002).Google Scholar
Doubrov, B., Generalised Wilczynski invariants for non-linear ordinary differential equations. Symmetries and overdetermined systems of partial differential equations, Eastwood, M., Miller, W. (eds.), IMA Vol. Math. Appl. 144 (Springer, NY 2008), 25–40.CrossRefGoogle Scholar
Doubrov, B., Ferapontov, E. V., Kruglikov, B. and Novikov, V. S.. On the integrability in Grassmann geometries: integrable systems associated with fourfolds in Gr(3, 5). Proc. London Math. Soc. (3) 116, no. 5 (2018), 1269–1300.Google Scholar
Dunajski, M. and Tod, P.. Paraconformal geometry of nth-order ODEs, and exotic holonomy in dimension four. J. Geom. Phys . 56 (2006), 1790–1809.CrossRefGoogle Scholar
Dunajski, M.. Interpolating integrable system. arXiv:0804.1234v1.Google Scholar
Dunajski, M. and Krynski, W.. Einstein–Weyl geometry, dispersionless Hirota equation and Veronese webs. Math. Proc. Camb Phil. Soc. 157, no. 1 (2014), 139–150.Google Scholar
Eisenbud, D.. Commutative Algebra with a View Toward Algebraic Geometry (Springer, Berlin 1995).Google Scholar
Ferapontov, E. V. and Khusnutdinova, K. R.. On the integrability of (2+1)-dimensional quasilinear systems. Comm. Math. Phys. 248 (2004), 187–206.CrossRefGoogle Scholar
Ferapontov, E. V., Hadjikos, L. and Khusnutdinova, K. R.. Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian. Int. Math. Res. Notices (2010), no. 3, 496–535.Google Scholar
Ferapontov, E. V. and Kruglikov, B.. Dispersionless integrable systems in 3D and Einstein-Weyl geometry. J. Diff. Geom. 97 (2014), 215–254.Google Scholar
Gabber, O.. The integrability of the characteristic variety. Amer. J. Math. 103, no. 3 (1981), 445–468.CrossRefGoogle Scholar
Gelfand, I. M. and Zakharevich, I.. Webs, Veronese curves, and bi-Hamiltonian systems. J. Funct. Anal. 99, no. 1 (1991), 150–178.CrossRefGoogle Scholar
Gindikin, S. G.. Generalised conformal structures. Twistors in mathematics and physics, 36-52, London Math. Soc. Lecture Note Ser. 156 (Cambridge university Press, Cambridge, 1990).Google Scholar
Godlinski, M. and Nurowski, P.. GL(2,R) geometry of ODE’s. J. Geom. Phys . 60, no. 6-8 (2010), 991–1027.CrossRefGoogle Scholar
Guillemin, V. W., Quillen, D. and Sternberg, S.. The integrability of characteristics. Comm. Pure Appl. Math. 23, no. 1 (1970), 39–77.CrossRefGoogle Scholar
Hartshorne, R.. Algebraic geometry, Graduate Texts in Mathematics, No. 52 (Springer-Verlag, New York-Heidelberg, 1977), 496 pp.CrossRefGoogle Scholar
Harris, J.. Algebraic Geometry: a First Course. (Springer-Verlag, New York 1992), 328pp.Google Scholar
Hitchin, N. J.. Complex manifolds and Einstein’s equations. Twistor geometry and nonlinear systems (Primorsko, 1980), 73–99, Lecture Notes in Math. 970, (Springer, Berlin-New York 1982).CrossRefGoogle Scholar
Konopelchenko, B. G. and Magri, F.. Dispersionless integrable equations as coisotropic deformations: generalizations and reductions. Theoret. and Math. Phys. 151, no. 3 (2007), 803–819.CrossRefGoogle Scholar
Konopelchenko, B. G. and Ortenzi, G.. Coisotropic deformations of algebraic varieties and integrable systems. J. Phys. A 42, no. 41 (2009), 415207, 18 pp.Google Scholar
Kruglikov, B. and Lychagin, V.. Geometry of Differential equations. Handbook of Global Analysis, Ed. Krupka, D., Saunders, D., (Elsevier, 2008) 725–772.Google Scholar
Kruglikov, B. and Lychagin, V.. Compatibility, multi-brackets and integrability of systems of PDEs. Acta Appl. Math . 109 (2010), 151–196.CrossRefGoogle Scholar
Kruglikov, B.. Lie theorem via rank 2 distributions (integration of PDE of class ω = 1). J. Nonlin. Math. Phys. 19, no. 2 (2012), 1250011.CrossRefGoogle Scholar
Kruglikov, B. and Panasyuk, A.. Veronese webs and nonlinear PDEs. J. Geom. Phys. 115 (2017), 45–60.Google Scholar
Krynski, W.. Paraconformal structures, ordinary differential equations and totally geodesic manifolds. J. Geom. Phys . 103 (2016), 1–19.CrossRefGoogle Scholar
Krynski, W. and Mettler, T.. GL(2, ℝ)-structures in dimension four, H-flatness and integrability, arXiv:1611.08228.Google Scholar
Kodaira, K.. On stability of compact submanifolds of complex manifolds. Amer. J. Math. 85 (1963), 79–94.CrossRefGoogle Scholar
Lie, S.. Zur allgemeinen teorie der partiellen differentialgleichungen beliebiger ordnung, Leipz. Berichte, Heft I, 53-128 (1895); Gesammelte Abhandlungen Bd.4, paper IX (Teubner-Aschehoung, Leipzig-Oslo, 1929).Google Scholar
Manakov, S. V. and Santini, P. M.. The Cauchy problem on the plane for the dispersionless Kadomtsev–Petviashvili equation. JETP Lett . 83 (2006), 462–6.CrossRefGoogle Scholar
Martinez Alonso, L. and Shabat, A. B.. Energy-dependent potentials revisited: a universal hierarchy of hydrodynamic type. Phys. Lett. A 300 (2002), 58–54.CrossRefGoogle Scholar
Merkulov, S. and Schwachhöfer, L.. Classification of irreducible holonomies of torsion-free affine connections. Ann. of Math. (2) 150, no. 1 (1999), 77–149.Google Scholar
Nurowski, P.. Differential equations and conformal structures. J. Geom. Phys. 55, no. 1 (2005), 19–49.CrossRefGoogle Scholar
Nurowski, P.. Comment on GL(2, R) geometry of fourth-order ODEs. J. Geom. Phys . 59, no. 3 (2009), 267–278.Google Scholar
Smith, A. D.. Integrable GL(2) geometry and hydrodynamic partial differential equations. Communications in Analysis and Geometry 18, no. 4 (2010), 743–790.CrossRefGoogle Scholar
Smith, A. D.. Involutive tableaux, characteristic varieties, and rank-one varieties in the geometric study of PDEs. Geometry of Lagrangian Grassmannians and nonlinear PDEs, 57-112, Banach Center Publ., 117, (Polish Acad. Sci. Inst. Math., Warsaw, 2019).CrossRefGoogle Scholar
Takasaki, K.. Auxiliary linear problem, difference Fay identities and dispersionless limit of Pfaff–Toda hierarchy. SIGMA Symmetry Integrability Geom. Methods Appl. 55 (2009), Paper 109, 34 pp.CrossRefGoogle Scholar
Tod, P.. Einstein-Weyl spaces and third-order differential equations, J. Math. Phys. 41, no. 8 (2000), 5572–5581.CrossRefGoogle Scholar
Ward, R. S.. Einstein-Weyl spaces and SU(∞) Toda fields. Class. Quantum Grav. 7, no. 4 (1990), L95–L98.CrossRefGoogle Scholar
Zakharevich, I.. Nonlinear wave equation, nonlinear Riemann problem, and the twistor transform of Veronese webs, arXiv:math-ph/0006001.Google Scholar