On (2+1)-dimensional hydrodynamic type systems possessing a pseudopotential with movable singularities



We consider a class of hydrodynamic type systems that have three independent and N ⩾ 2 dependent variables and possess a pseudopotential. It turns out that systems having a pseudopotential with movable singularities can be described by some functional equation. We find all solutions of this equation, which permits constructing interesting examples of integrable systems of hydrodynamic type for arbitrary N.

Key words

integrable (2+1)-dimensional hydrodynamic type system pseudopotential with movable singularities 


  1. [1]
    E. V. Ferapontov and K. R. Khusnutdinova, “Hydrodynamic reductions of multi-dimensional dispersionless PDEs: the test for integrability,” J. Math. Phys., 45:6 (2004), 2365–2377.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    E. V. Ferapontov and K. R. Khusnutdinova, “The characterization of two-component (2+1)-dimensional integrable systems of hydrodynamic type,” J. Phys. A: Math. Gen., 37:8 (2004), 2949–2963.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    E. V. Ferapontov and K. R. Khusnutdinova “The Haantjes tensor and double waves for multidimensional systems of hydrodynamic type: a necessary condition for integrability,” Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462:2068 (2006), 1197–1219.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    J. Gibbons and S. P. Tsarev, “Reductions of the Benney equations,” Phys. Lett. A, 211:1 (1996), 19–24.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    I. M. Krichever, “Method of averaging for two-dimensional “integrable” equations,” Funkts. Anal. Prilozhen., 22:3 (1988), 37–52; English transl.: Functional Anal. Appl., 22:3 (1988), 200–213.MathSciNetGoogle Scholar
  6. [6]
    I. M. Krichever, “The dispersionless Lax equations and topological minimal models,” Comm. Math. Phys., 143:2 (1992), 415–429.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    I. M. Krichever, “The τ-function of the universal Whitham hierarchy, matrix models and topological field theories,” Comm. Pure Appl. Math., 47:4 (1994), 437–475.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    V. E. Zakharov, “Dispersionless limit of integrable systems in 2+1 dimensions,” in: Singular Limits of Dispersive Waves, Plenum Press, New York, 1994, 165–174.Google Scholar
  9. [9]
    S. P. Tsarev, “The geometry of Hamiltonian systems of hydrodynamic type. Generalized hodograph method,” Izv. Akad. Nauk SSSR, Ser. Math., 54:5 (1990), 1048–1068; English transl.: Math. USSR-Izv., 37:2 (1991), 397–419.Google Scholar
  10. [10]
    S. V. Manakov and P. M. Santini, “Cauchy problem on the plane for the dispersionless Kadomtsev-Petviashvili equation,” JETP Lett., 83:10 (2006), 462–466.CrossRefGoogle Scholar
  11. [11]
    S. V. Manakov and P. M. Santini, “Inverse scattering problem for vector fields and the Cauchy problem for the heavenly equation,” Phys. Lett. A, 359:6 (2006), 613–619.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    J. Haantjes, “On X m-forming sets of eigenvectors,” Indag. Math., 17 (1955), 158–162.MathSciNetGoogle Scholar
  13. [13]
    Z. Peradzy’nski, “Nonlinear planar k-waves and Riemann invariants,” Bull. Acad. Polon. Sci. Sr. Sci. Tech., 19 (1971), 625–632.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Landau Institute for Theoretical PhysicsMoscowRussia
  2. 2.School of MathematicsThe University of ManchesterMoscowRussia

Personalised recommendations