Integrable Equations of the Dispersionless Hirota type and Hypersurfaces in the Lagrangian Grassmannian

International Mathematics Research Notices - Tập 2010 Số 3 - Trang 496-535 - 2010
E. V. Ferapontov1, Lenos Hadjikos1, К. Р. Хуснутдинова1
1Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, United Kingdom

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Ablowitz, 2006, “Integrable systems and modular forms of level 2”, Journal of Physics A: Mathematical and General, 39, 15341, 10.1088/0305-4470/39/50/003

Atiyah, 1978, “Self-duality in four-dimensional Riemannian geometry”, Proceedings of the Royal Society of London, Series, 425, 10.1098/rspa.1978.0143

Akivis, 1999, “Semiintegrable almost Grassmann structures”, Differential Geometry and Its Applications, 10, 257, 10.1016/S0926-2245(99)00014-5

Bailey, 1991, “Complex paraconformal manifolds – their differential geometry and twistor theory”, Forum Mathematicum, 3, 61, 10.1515/form.1991.3.61

Banos, 2003, “On symplectic classification of effective 3-forms and Monge-Ampére equations”, Differential Geometry and Its Applications, 19, 147, 10.1016/S0926-2245(03)00017-2

Bertram, 1996, “Un theoréme de Liouville pour les algébres de Jordan”, Bulletin de la Société Mathématique de France, 124, 299, 10.24033/bsmf.2282

Bobienski, 2007, “Irreducible SO(3) geometry in dimension five”, Journal für die Reine und Angewandte Mathematik, 605, 51

Bogdanov, 2005, “On dispersionless BKP hierarchy and its reductions”, Journal of Nonlinear Mathematical Physics, 12, 64, 10.2991/jnmp.2005.12.s1.6

Boyarsky, 2001, “Associativity equations in dispersionless integrable hierarchies”, Physics Letters, B 515, 483, 10.1016/S0370-2693(01)00893-0

Boyer, 1982, “Killing vectors in self-dual Euclidean Einstein spaces”, Journal of Mathematical Physics, 23, 1126, 10.1063/1.525479

Braden, 2002, “WDVV equations as functional relations”, Physics Letters, B 451, 376, 10.1016/S0370-2693(02)02225-6

Buchstaber, 2003, “The Egorov hydrodynamic chains, the Chazy equation and SL(2, C)group”, Functional Analysis and Its Applications, 37, 13, 10.1023/B:FAIA.0000015576.05085.bc

Burovskii, 2008, “Second order quasilinear PDEs and conformal structures in projective space”

Calabi, 1958, “Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jorgens”, Michigan Mathematical Journal, 5, 105, 10.1307/mmj/1028998055

Carroll, 1995, “Solution of the dispersionless Hirota equations”, Journal of Physics, A 28, 6373

Cartan, 1909, “Les groupes de transformations continus, infinis, simples”, Annales Scientifiques de l'École Normale Supérieure, 26, 93, 10.24033/asens.603

Doubrov, “Contact Trivialization of Ordinary Differential Equations”, Differential Geometry and Its Applications, 73

Dunajski, 2001, “Einstein-Weyl geometry, the dKP equation and twistor theory”, Journal of Geometry and Physics, 37, 63, 10.1016/S0393-0440(00)00033-4

Dunajski, 2006, “Paraconformal geometry of nth-order ODEs, and exotic holonomy in dimension four”, Journal of Geometry and Physics, 56, 1790, 10.1016/j.geomphys.2005.10.007

Ferapontov, 2004, “On the integrability of (2+1)-dimensional quasilinear systems”, Communications in Mathematical Physics, 248, 187, 10.1007/s00220-004-1079-6

Ferapontov, 2004, “The characterization of two-component (2+1)-dimensional integrable systems of hydrodynamic type”, Journal of Physics A: Mathematical and General, 37, 2949, 10.1088/0305-4470/37/8/007

Ferapontov, 2002, “Boyer-Finley equation and systems of hydrodynamic type”, Classical and Quantum Gravity, 19, L205, 10.1088/0264-9381/19/24/101

Ferapontov, 2003, “Hydrodynamic reductions of the heavenly equation”, Classical and Quantum Gravity, 20, 2429, 10.1088/0264-9381/20/11/331

Ferapontov, 2004, “Hydrodynamic reductions of multi-dimensional dispersionless PDEs: the test for integrability”, Journal of Mathematical Physics, 45, 2365, 10.1063/1.1738951

Ferapontov, 2005, “Classification of integrable (2+1)-dimensional quasilinear hierarchies”, Theoretical and Mathematical Physics, 144, 35, 10.1007/s11232-005-0117-7

Gibbons, 1996, “Reductions of the Benney equations”, Physics Letters, A 211, 19, 10.1016/0375-9601(95)00954-X

Gibbons, 1999, “Conformal maps and reductions of the Benney equations”, Physics Letters, A 258, 263, 10.1016/S0375-9601(99)00389-8

Gindikin, 1998, “On the automorphism group of the generalized conformal structure of a symmetric R-space”, Differential Geomeometry and Its Applications, 8, 21, 10.1016/S0926-2245(97)00015-6

Grant, 1993, “On self-dual gravity”, Physical Review, D 47, 2606

Harris, 1992, “Algebraic geometry: A first course”

Hoevenaars, 2005, “Second order reductions of the WDVV equations related to classical Lie algebras”, Letters in Mathematical Physics, 71, 83, 10.1007/s11005-004-6029-x

Ibragimov, Transformation Groups Applied to Mathematical Physics

Joyce, 2005, Lectures on Special Lagrangian Geometry

Krichever, 1994, “The τ-function of the universal Whitham hierarchy, matrix models and topological field theories”, Communications on Pure and Applied Mathematics, 47, 437, 10.1002/cpa.3160470403

Krichever, 2005, “Integrable structure of the Dirichlet boundary problem in multiply-connected domains”, Communications in Mathematical Physics, 259, 1, 10.1007/s00220-005-1387-5

Landsberg, 2001, “The projective geometry of Freudental's magic square”, Journal of Algebra, 239, 477, 10.1006/jabr.2000.8697

Landsberg, 2002, “Construction and classification of complex simple Lie algebras via projective geometry”, Selecta Mathematica, New Series, 8, 137, 10.1007/s00029-002-8103-5

Lychagin, 1993, “A classification of Monge-Amp.re equations”, Annales Scientifiques de l'École Normale Supérieure, 26, 281, 10.24033/asens.1673

Beffa, 2007, “On completely integrable geometric evolutions of curves of Lagrangian planes”, Proc. Roy. Soc. Edinburgh Sect. A, 137, 111, 10.1017/S0308210505000600

Mañas, 2002, “Reductions and hodograph solutions of the dispersionless KP hierarchy”, Journal of Physics, A 35, 401

Marshakov, 2002, “Integrable structure of the Dirichlet boundary problem in two dimensions”, Communications in Mathematical Physics, 227, 131, 10.1007/s002200200629

Mokhov, 1994, “Bianchi transformation between the real hyperbolic Monge-Ampére equation and the Born Infeld equation”, Letters in Mathematical Physics, 32, 121, 10.1007/BF00739421

Mokhov, 2001, Symplectic and Poisson Geometry on Loop Spaces of Smooth Manifolds and Integrable Equations

Mukai, “Simple Lie algebras and Legendre variety”

Olver, 1986, Applications of Lie Groups to Differential Equations, 10.1007/978-1-4684-0274-2

Ovsienko, 1993, “Lagrange Schwarzian derivative and symplectic Sturm theory”, Annales de la Faculté des Sciences de Toulouse, 2, 73, 10.5802/afst.758

Pavlov, 2003, “New integrable (2+1)-equations of hydrodynamic type”, Russian Mathematical Surveys, 58, 384, 10.1070/RM2003v058n02ABEH000620

Pavlov, 2004, “Classifying integrable Egoroff hydrodynamic chains”, Theoretical and Mathematical Physics, 138, 45, 10.1023/B:TAMP.0000010632.20218.62

Pinkall, 1985, “Dupinishe Hyperflachen in E4”, Manuscripta Mathematica, 51, 89, 10.1007/BF01168348

Plebański, 1975, “Some solutions of complex Einstein equations”, Journal of Mathematical Physics, 16, 2395, 10.1063/1.522505

Plebański, 1996, “The Lagrangian for a self-dual gravitational field as a limit of the SDYM Lagrangian”, Physics Letters, A 212, 22, 10.1016/0375-9601(96)00025-4

Takasaki, 1995, “Integrable hierarchies and dispersionless limit”, Reviews in Mathematical Physics, 7, 743, 10.1142/S0129055X9500030X

Tsarev, 1985, “Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type”, Soviet Mathematics: Doklady, 31, 488

Tsarev, 1991, “The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method”, Izvestiya Mathematics of the USSR: Izvestiya 37, 397, 10.1070/IM1991v037n02ABEH002069

Wiegmann, 2000, “Conformal maps and integrable hierarchies”, Communications in Mathematical Physics, 213, 523, 10.1007/s002200000249

Zabolotskaya, 1969, “Quasi-plane waves in the nonlinear acoustics of confined beams”, Soviet Physics: Acoustics, 15, 35

Zabrodin, 2001, “The dispersionless limit of the Hirota equations in some problems of complex analysis”, Theoretical and Mathematical Physics, 129, 1511, 10.1023/A:1012883123413

Zakharov, 1994, “Dispersionless limit of integrable systems in 2 + 1dimensions”, Singular Limits of Dispersive Waves, 165, 10.1007/978-1-4615-2474-8_13

Zelenko, 2005, “Complete Systems of Invariants for Rank 1 Curves in Lagrange Grassmannians”, Differential Geometry and Its Applications, 367