Advertisement

Mathematische Annalen

, Volume 339, Issue 1, pp 61–99 | Cite as

Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor

  • E. V. Ferapontov
  • D. G. Marshall
Article

Abstract

The integrability of an m-component system of hydrodynamic type, u t = V(u)u x , by the generalized hodograph method requires the diagonalizability of the m ×  m matrix V(u). This condition is known to be equivalent to the vanishing of the corresponding Haantjes tensor. We generalize this approach to hydrodynamic chains—infinite-component systems of hydrodynamic type for which the ∞ ×  ∞ matrix V(u) is ‘sufficiently sparse’. For such systems the Haantjes tensor is well-defined, and the calculation of its components involves finite summations only. We illustrate our approach by classifying broad classes of conservative and Hamiltonian hydrodynamic chains with the zero Haantjes tensor. We prove that the vanishing of the Haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semi-Hamiltonian hydrodynamic reductions, thus providing an easy-to-verify necessary condition for the integrability.

Mathematics Subject Classification (2000)

35L40 35L65 37K10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alber S.J. (1991). Associated integrable systems. J. Math. Phys. 32: 916–922 MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Benney D.J. (1973). Some properties of long nonlinear waves. Stud. Appl. Math. 52: 45–50 MATHGoogle Scholar
  3. 3.
    Blaszak M. (2002). Classical R-matrices on Poisson algebras and related dispersionless systems. Phys. Lett. A 297(3–4): 191–195 MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bukhshtaber V.M., Leikin D.V. and Pavlov M.V. (2003). Egorov hydrodynamic chains, the Chazy equation, and the group SL (2, C). Funktsional Anal. Prilozhen. 37(4): 13–26 MathSciNetGoogle Scholar
  5. 5.
    Chazy J. (1910). Sur les équations différentielles dont l’intégrale générale poss ède un coupure essentielle mobile. C.R. Acad. Sci. Paris 150: 456–458MATHGoogle Scholar
  6. 6.
    Dubrovin B.A. and Novikov S.P. (1989). Hydrodynamics of weakly deformed soliton lattices: differential geometry and Hamiltonian theory. Russ. Math. Surv. 44(6): 35–124 MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dubrovin, B.A.: Geometry of 2D topological field theories. Lect. Notes in Math, 1620. Springer, Heidelberg, 120–348 (1996)Google Scholar
  8. 8.
    Dubrovin B.A., Liu S. and Zhang Y. (2006). On Hamiltonian perturbations of hyperbolic systems of conservation laws. I. Quasi-triviality of bi-Hamiltonian perturbations. Comm. Pure Appl. Math. 59(4): 559–615 MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Ferapontov E.V. (1995). Nonlocal Hamiltonian operators of hydrodynamic type: differential geometry and applications. Am. Math. Soc. Transl. 170(2): 33–58 MathSciNetGoogle Scholar
  10. 10.
    Ferapontov E.V. and Khusnutdinova K.R. (2004). On integrability of (2+1)-dimensional quasilinear systems. Comm. Math. Phys. 248: 187–206 arXiv:nlin.SI/0305044MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Ferapontov E.V. and Khusnutdinova K.R. (2004). The characterization of two-component (2+1)-dimensional integrable systems of hydrodynamic type. J. Phys. A Math. Gen. 37(N8): 2949–2963 arXiv:nlin. SI/0310021MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Ferapontov E.V. and Khusnutdinova K.R. (2004). Hydrodynamic reductions of multi-dimensional dispersionless PDEs: the test for integrability. J. Math. Phys. 45(N6): 2365–2377 arXiv:nlin.SI/0312015MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ferapontov E.V. and Khusnutdinova K.R. (2006). Double waves in multi-dimensional systems of hydrodynamic type: the necessary condition for integrability. Proc. R. Soc. A 462: 1197–1219 arXiv:nlin. SI/0412064CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Ferapontov E.V., Khusnutdinova K.R. and Pavlov M.V. (2005). Classification of integrable (2+1)-dimensional quasilinear hierarchies. Theor. Math. Phys. 144: 35–43 CrossRefMathSciNetGoogle Scholar
  15. 15.
    Ferapontov, E.V., Khusnutdinova, K.R., Marshall, D.G., Pavlov, M.V.: Classification of integrable Hamiltonian hydrodynamic chains associated with Kupershmidt’s brackets. J. Math. Phys. 47(1) (2006)Google Scholar
  16. 16.
    Ferapontov E.V., Khusnutdinova K.R. and Tsarev S.P. (2006). On a class of three-dimensional integrable Lagrangians. Comm. Math. Phys. 261(N1): 225–243 MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Gibbons J. (1981). Collisionless Boltzmann equations and integrable moment equations. Physica D 3: 503–511 CrossRefMathSciNetGoogle Scholar
  18. 18.
    Gibbons J. and Kodama Y. (1989). A method for solving the dispersionless KP hierarchy and its exact solutions II. Phys. Lett. A 135(3): 167–170 CrossRefMathSciNetGoogle Scholar
  19. 19.
    Gibbons J. and Tsarev S.P. (1996). Reductions of the Benney equations. Phys. Lett. A 211: 19–24 MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Gibbons J. and Tsarev S.P. (1999). Conformal maps and reductions of the Benney equations. Phys. Lett. A 258: 263–271 MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Gibbons, J., Raimondo, A.: Differential geometry of hydrodynamic Vlasov equations, arXiv:nlin.SI/ 0612022Google Scholar
  22. 22.
    Haantjes J. (1955). On X m-forming sets of eigenvectors. Indagationes Math. 17: 158–162 MathSciNetGoogle Scholar
  23. 23.
    Konopelchenko B.G., Martinez Alonso L., Ragnisco O. (2001) The \({\overline\partial}\) -approach to the dispersionless KP hierarchy. J. Phys. A 34(47): 10209–10217Google Scholar
  24. 24.
    Krichever I.M. (1994). The τ-function of the universal Whitham hierarchy, matrix models and topological field theorie. Comm. Pure Appl. Math. 47(4): 437–475 MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Kupershmidt B.A. and Manin Yu.I. (1977). Long wave equations with a free surface. I. Conservation laws and solutions (Russian). Funktsional. Anal. Prilozhen. 11(3): 31–42 MATHMathSciNetGoogle Scholar
  26. 26.
    Kupershmidt B.A. and Manin Yu.I. (1978). Long wave equations with a free surface. II. The Hamiltonian structure and the higher equations (Russian). SFunktsional. Anal. Prilozhen. 12(1): 25–37 MATHMathSciNetGoogle Scholar
  27. 27.
    Kupershmidt B.A. (1983). Deformations of integrable systems. Proc. Roy. Irish Acad. Sect. A 83(1): 45–74 MathSciNetMATHGoogle Scholar
  28. 28.
    Kupershmidt B.A. (1990). The quasiclassical limit of the modified KP hierarchy. J. Phys. A: Math. Gen 23: 871–886 MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Kupershmidt, B.A.: Normal and universal forms in integrable hydrodynamical systems. In: Proceedings of the Berkeley–Ames conference on nonlinear problems in control and fluid dynamics (Berkeley, Calif., 1983), pp 357–378, in Lie Groups: Hist. Frontiers and Appl Ser. B: Systems Inform. Control, II, Math Sci Press, Brookline (1984)Google Scholar
  30. 30.
    Kupershmidt, B.A.: Extended equations of long waves, preprint (2005)Google Scholar
  31. 31.
    Lebedev D.R. and Manin Yu.I. (1979). Conservation laws and Lax representation for Benney’s long wave equations. Phys. lett. A 74: 154–156 CrossRefMathSciNetGoogle Scholar
  32. 32.
    Mañas M. (2004). On the rth dispersionless Toda hierarchy: factorization problem, additional symmetries and some solutions. J. Phys. A Math. Gen. 37: 9195–9224 MATHCrossRefGoogle Scholar
  33. 33.
    Mañas M. (2004). S-functions, reductions and hodograph solutions of the rth dispersionless modified KP and Dym hierarchies. J. Phys. A Math. Gen. 37: 11191–11221 MATHCrossRefGoogle Scholar
  34. 34.
    Mañas M., Martinez Alonso L. and Medina E. (2002). Reductions and hodograph solutions of the dispersionless KP hierarchy. J. Phys. A Math. Gen. 35: 401–417 MATHCrossRefGoogle Scholar
  35. 35.
    Mikhalev, V.G.: Hamiltonian formalism of Korteweg-de Vries-type hierarchies (Russian). Funktsional. Anal. i Prilozhen. 26(2) 79–82 (1992); translation in Funct. Anal. Appl. 26(2) 140–142 (1992)Google Scholar
  36. 36.
    Miura R.M. (1974). Conservation laws for fully nonlinear long wave equations. Stud. Appl. Math. 53: 45–56 MathSciNetMATHGoogle Scholar
  37. 37.
    Pavlov M.V. (2003). Integrable hydrodynamic chains. J. Math. Phys. 44(9): 4134–4156 MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Pavlov M.V. (2004). Classification of integrable Egorov hydrodynamic chains. Teoret. Mat. Fiz. 138(1): 55–70 MathSciNetGoogle Scholar
  39. 39.
    Pavlov M.V., Svinolupov S.I. and Sharipov R.A. (1996). An invariant criterion for hydrodynamic integrability. Funktsional Anal. Prilozhen. 30: 18–29 Translation in Funct Anal Appl 301522MathSciNetGoogle Scholar
  40. 40.
    Pavlov, M.V.: Modified dispersionless Veselov–Novikov equations and corresponding hydrodynamic chains, arXiv:nlin.SI/0611022Google Scholar
  41. 41.
    Pavlov, M.V.: The Hamiltonian approach in classification and integrability of hydrodynamic chains, arXiv:nlin.SI/0603057Google Scholar
  42. 42.
    Peradzyński Z. (1971). Nonlinear plane k-waves and Riemann invariants. Bull. Acad. Polon. Sci. Sr. Sci. Tech. 19: 625–632 MATHGoogle Scholar
  43. 43.
    Sévennec, B.: Géométrie des systèmes hyperboliques de lois de conservation, Mémoire (nouvelle série) N56, Supplément au Bulletin de la Société Mathématique de France 122, 1–125 (1994)Google Scholar
  44. 44.
    Martinez Alonso L. and Shabat A.B. (2003). Towards a theory of differential constraints of a hydrodynamic hierarchy. J. Nonlinear Math. Phys. 10(2): 229–242 MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Martinez Alonso, L., Shabat, A.B.: Energy-dependent potentials revisited: a universal hierarchy of hydrodynamic type. Phys. lett. A 299(4), 359–365 (2002); Phys. Lett. A 300(1), 58–64 (2002)Google Scholar
  46. 46.
    Teshukov V.M. (1985). On the hyperbolicity of long wave equations. Dokl. Akad. Nauk SSSR 284(3): 555–559 MathSciNetGoogle Scholar
  47. 47.
    Teshukov, V.M.: Characteristics, conservation laws and symmetries of the kinetic equations of motion of bubbles in a fluid. Prikl. Mekh. Tekhn. Fiz. 40(2), (1999) 86–100; translation in J. Appl. Mech. Tech. Phys. 40(2), 263–275 (1999)Google Scholar
  48. 48.
    Tsarev S.P. (1990). Geometry of Hamiltonian systems of hydrodynamic type. Generalized hodograph method. Izvestija AN USSR Math. 54(5): 1048–1068 Google Scholar
  49. 49.
    Zakharov V.E. (1980). Benney equations and quasiclassical approximation in the inverse problem method. Funktsional Anal. i Prilozhen. 14(2): 15–24 MATHMathSciNetCrossRefGoogle Scholar
  50. 50.
    Zakharov V.E. (1981). On the Benney equations. Physica D 3(1,2): 193–202 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of Mathematical SciencesLoughborough UniversityLoughboroughUK

Personalised recommendations