Journal of Applied Mathematics

Compatible flat metrics

Oleg I. Mokhov

Full-text: Open access

Abstract

We solve the problem of description of nonsingular pairs of compatible flat metrics for the general N-component case. The integrable nonlinear partial differential equations describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) are found and integrated. The integrating of these equations is based on reducing to a special nonlinear differential reduction of the Lamé equations and using the Zakharov method of differential reductions in the dressing method (a version of the inverse scattering method).

Article information

Source
J. Appl. Math., Volume 2, Number 7 (2002), 337-370.

Dates
First available in Project Euclid: 30 March 2003

Permanent link to this document
https://projecteuclid.org/euclid.jam/1049074868

Digital Object Identifier
doi:10.1155/S1110757X02203149

Mathematical Reviews number (MathSciNet)
MR1942027

Zentralblatt MATH identifier
1008.37041

Subjects
Primary: 37K10: Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.) 37K15: Integration of completely integrable systems by inverse spectral and scattering methods
Secondary: 37K25: Relations with differential geometry 35Q58 53B20: Local Riemannian geometry 53B21: Methods of Riemannian geometry 53B50: Applications to physics 53A45: Vector and tensor analysis

Citation

Mokhov, Oleg I. Compatible flat metrics. J. Appl. Math. 2 (2002), no. 7, 337--370. doi:10.1155/S1110757X02203149. https://projecteuclid.org/euclid.jam/1049074868


Export citation

References

  • M. Arik, F. Neyzi, Y. Nutku, P. J. Olver, and J. M. Verosky, Multi-Hamiltonian structure of the Born-Infeld equation, J. Math. Phys. 30 (1989), no. 6, 1338–1344.
  • D. B. Cooke, Classification results and the Darboux theorem for low-order Hamiltonian operators, J. Math. Phys. 32 (1991), no. 1, 109–119.
  • ––––, Compatibility conditions for Hamiltonian pairs, J. Math. Phys. 32 (1991), no. 11, 3071–3076.
  • G. Darboux, Leçons sur les Systèmes Orthogonaux et les Coordonnées Curvilignes, 2nd ed., Gauthier-Villars, Paris, 1910 (French).
  • I. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations, John Wiley & Sons, Chichester, 1993.
  • B. Dubrovin, Geometry of $2$D topological field theories, Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lecture Notes in Mathematics, vol. 1620, Springer, Berlin, 1996, pp. 120–348.
  • information of [7,6,8?] according to MathSciNet database. Please check.} of orbits of a Coxeter group}}, Surveys in Differential Geometry: Integral Systems [Integrable Systems], Surveys in Differential Geometry, vol. 4, International Press, Massachusetts, 1998, pp. 181–211.
  • ––––, Flat pencils of metrics and Frobenius manifolds, Integrable Systems and Algebraic Geometry (Kobe/Kyoto, 1997), World Scientific Publishing, New Jersey, 1998, pp. 47–72.
  • B. Dubrovin and S. P. Novikov, Hamiltonian formalism of one-dimensional systems of the hydrodynamic type and the Bogolyubov-Whitham averaging method, Dokl. Akad. Nauk SSSR 270 (1983), no. 4, 781–785, translated in Soviet Math. Dokl. 27 (1983), 665–669.
  • ––––, Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory, Uspekhi Mat. Nauk 44 (1989), no. 6, 29–98, translated in Russian Math. Surveys 44 (1989), no. 6, 35–124.
  • E. V. Ferapontov, Hamiltonian systems of hydrodynamic type and their realizations on hypersurfaces of a pseudo-Euclidean space, Problems in Geometry, Vol. 22 (Russian), Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990, translated in J. Soviet Math. 55 (1991), no. 5, 1970–1995, pp. 59–96.
  • ––––, Differential geometry of nonlocal Hamiltonian operators of hydrodynamic type, Funktsional. Anal. i Prilozhen. 25 (1991), no. 3, 37–49, translated in Funct. Anal. Appl. 25 (1991), no. 3, 195–204.
  • ––––, Nonlocal Hamiltonian operators of hydrodynamic type: differential geometry and applications, Topics in Topology and Mathematical Physics (S. P. Novikov, ed.), Amer. Math. Soc. Transl. Ser. 2, vol. 170, American Mathematical Society, Rhode Island, 1995, pp. 33–58.
  • E. V. Ferapontov and M. V. Pavlov, Quasiclassical limit of coupled KdV equations. Riemann invariants and multi-Hamiltonian structure, Phys. D 52 (1991), no. 2-3, 211–219.
  • A. S. Fokas and B. Fuchssteiner, On the structure of symplectic operators and hereditary symmetries, Lett. Nuovo Cimento (2) 28 (1980), no. 8, 299–303.
  • B. Fuchssteiner, Application of hereditary symmetries to nonlinear evolution equations, Nonlinear Anal. 3 (1979), no. 6, 849–862.
  • C. S. Gardner, Korteweg-de Vries equation and generalizations. IV. The Korteweg-de Vries equation as a Hamiltonian system, J. Mathematical Phys. 12 (1971), 1548–1551.
  • I. M. Gel'fand and I. Dorfman, Hamiltonian operators and algebraic structures associated with them, Funktsional. Anal. i Prilozhen. 13 (1979), no. 4, 13–30, translated in Funct. Anal. Appl. 13 (1979), 246–262.
  • H. Gümral and Y. Nutku, Multi-Hamiltonian structure of equations of hydrodynamic type, J. Math. Phys. 31 (1990), no. 11, 2606–2611.
  • “$X_{n-1}$” to “${X}\sb m$” in [20?] according to MathSciNet database. Please check.} sets of eigenvectors}}, Indag. Math. 17 (1955), 158–162.
  • I. M. Krichever, Algebraic-geometric $n$-orthogonal curvilinear coordinate systems and the solution of associativity equations, Funktsional. Anal. i Prilozhen. 31 (1997), no. 1, 32–50, translated in Funct. Anal. Appl. 31 (1997), no. 1, 25–39.
  • F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), no. 5, 1156–1162.
  • O. I. Mokhov, Local third-order Poisson brackets, Uspekhi Mat. Nauk 40 (1985), no. 5, 257–258, translated in Russian Math. Surveys 40 (1985), 233–234.
  • ––––, Hamiltonian differential operators and contact geometry, Funktsional. Anal. i Prilozhen. 21 (1987), no. 3, 53–60, translated in Funct. Anal. Appl. 21 (1987), 217–223.
  • ––––, On compatible Poisson structures of hydrodynamic type, Uspekhi Mat. Nauk 52 (1997), no. 6, 171–172, translated in Russian Math. Surveys 52 (1997), no. 6, 1310–1311. \CMP1+611+3141 611 314
  • ––––, On compatible potential deformations of Frobenius algebras and associativity equations, Uspekhi Mat. Nauk 53 (1998), no. 2, 153–154, translated in Russian Math. Surveys 53 (1998), no. 2, 396–397.
  • ––––, Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems, Uspekhi Mat. Nauk 53 (1998), no. 3, 85–192, translated in Russian Math. Surveys 53 (1998), no. 3, 515–622.
  • ––––, Compatible Poisson structures of hydrodynamic type and associativity equations, Tr. Mat. Inst. Steklova 225 (1999), 284–300, translated in Proc. Steklov Inst. Math. 225 (1999), no. 2, 269–284.
  • ––––, Compatible Poisson structures of hydrodynamic type and the equations of associativity in two-dimensional topological field theory, Rep. Math. Phys. 43 (1999), no. 1-2, 247–256.
  • O. I. Mokhov and E. V. Ferapontov, Nonlocal Hamiltonian operators of hydrodynamic type that are connected with metrics of constant curvature, Uspekhi Mat. Nauk 45 (1990), no. 3(273), 191–192, translated in Russian Math. Surveys 45 (1990), no. 3, 218–219.
  • F. Neyzi, Diagonalization and Hamiltonian structures of hyperbolic systems, J. Math. Phys. 30 (1989), no. 8, 1695–1698.
  • A. Nijenhuis, ${X}\sb {n-1}$-forming sets of eigenvectors, Indag. Math. 13 (1951), 200–212.
  • Y. Nutku, On a new class of completely integrable nonlinear wave equations. II. Multi-Hamiltonian structure, J. Math. Phys. 28 (1987), no. 11, 2579–2585.
  • P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1986.
  • P. J. Olver and Y. Nutku, Hamiltonian structures for systems of hyperbolic conservation laws, J. Math. Phys. 29 (1988), no. 7, 1610–1619.
  • V. E. Zakharov, Description of the $n$-orthogonal curvilinear coordinate systems and Hamiltonian integrable systems of hydrodynamic type. I. Integration of the Lamé equations, Duke Math. J. 94 (1998), no. 1, 103–139.
  • V. E. Zakharov and L. D. Faddeev, The Korteweg-de Vries equation according to MathSciNet database. Please check. integrable Hamiltonian system}, Funkcional. Anal. i Priložen. 5 (1971), no. 4, 18–27, translated in Funct. Anal. Appl. 5 (1971), 280–287.