A sufficient condition for a rational differential operator to generate an integrable system

Abstract

For a rational differential operator \({L=AB^{-1}}\), the Lenard–Magri scheme of integrability is a sequence of functions \({F_n, n \geq 0}\), such that (1) \({B(F_{n+1})=A(F_n)}\) for all \({n \geq 0}\) and (2) the functions \({B(F_n)}\) pairwise commute. We show that, assuming that property (1) holds and that the set of differential orders of \({B(F_n)}\) is unbounded, property (2) holds if and only if L belongs to a class of rational operators that we call integrable. If we assume moreover that the rational operator L is weakly non-local and preserves a certain splitting of the algebra of functions into even and odd parts, we show that one can always find such a sequence (F n ) starting from any function in Ker B. This result gives some insight in the mechanism of recursion operators, which encode the hierarchies of the corresponding integrable equations.

This is a preview of subscription content, log in to check access.

References

  1. BDSK09

    Barakat A., De Sole A., Kac V.G.: Poisson vertex algebras in the theory of Hamiltonian equations. Jpn. J. Math. 4, 141–252 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  2. CDSK12

    Carpentier S., De Sole A., Kac V.G.: Some algebraic properties of differential operators. J. Math. Phys. 53, 063501 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  3. CDSK13

    Carpentier S., De Sole A., Kac V.G.: Some remarks on non-commutative principal ideal rings. C. R. Math. Acad. Sci. Paris 351, 5–8 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  4. CDSK14

    Carpentier S., De Sole A., Kac V.G.: Rational matrix pseudodifferential operators. Selecta Math. (N.S.) 20, 403–419 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  5. DS08

    Demskoi D.K., Sokolov V.V.: On recursion operators for elliptic models. Nonlinearity 21, 1253–1264 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  6. DSK13

    De Sole A., Kac V.G.: Non-local Poisson structures and applications to the theory of integrable systems. Jpn. J. Math. 8, 233–347 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  7. DSKT15

    De Sole A., Kac V.G., Turhan R.: On integrability of some bi-Hamiltonian two field systems of partial differential equations. J. Math. Phys. 56, 051503 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  8. GD76

    Gel’fand I.M., Dikii L.A.: A Lie algebra structure in a formal variational calculation. Funct. Anal. Appl. 10, 16–22 (1976)

    Article  MATH  Google Scholar 

  9. EOR93

    Enriquez B., Rubtsov V., Orlov A.: Higher Hamiltonian structures (the sl 2 case). JETP Lett. 58, 658–664 (1993)

    MathSciNet  Google Scholar 

  10. IS80

    Ibragimov N.H., Shabat A.B.: Evolution equations with a nontrivial Lie-Bäcklund group. Funktsional. Anal. i Prilozhen. 14(no. 1), 25–36 (1980)

    MathSciNet  Article  MATH  Google Scholar 

  11. MN01

    Maltsev A.Y., Novikov S.P.: On the local systems Hamiltonian in the weakly non-local Poisson brackets. Phys. D 156, 53–80 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  12. MS08

    A.V. Mikhailov and V.V. Sokolov, Symmetries of differential equations and the problem of integrability, In: Integrability, (ed. A.V. Mikhailov), Lecture Notes in Phys., 767, Springer-Verlag, 2009, pp. 19–88.

  13. MSS91

    A.V. Mikhailov, A.B. Shabat and V.V. Sokolov, The symmetry approach to classification of integrable equations, In: What is Integrability?, (ed. V.E. Zakharov), Springer-Verlag, 1991, pp. 115–184.

  14. Olv93

    P.J. Olver, Applications of Lie Groups to Differential Equations. Second ed., Springer-Verlag, 1993.

  15. SS84

    V.V. Sokolov and A.B. Shabat, Classification of integrable evolution equations, In: Mathematical Physics Reviews, (ed. S.P. Novikov), Soviet Sci. Rev. Sect. C Math. Phys. Rev., 4, Harwood Academic Publ., Chur, 1984, pp. 221–280.

  16. SW09

    J.A. Sanders and J.P. Wang, Number theory and the symmetry classification of integrable systems, In: Integrability, (ed. A.V. Mikhailov), Lecture Notes in Phys., 767, Springer-Verlag, 2009, pp. 89–118.

  17. Wan02

    Wang J.P.: A list of 1 + 1 dimensional integrable equations and their properties. J. Nonlinear Math. Phys. 9(suppl. 1), 213–233 (2002)

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Sylvain Carpentier.

Additional information

Communicated by: Yasuyuki Kawahigashi

About this article

Verify currency and authenticity via CrossMark

Cite this article

Carpentier, S. A sufficient condition for a rational differential operator to generate an integrable system. Jpn. J. Math. 12, 33–89 (2017). https://doi.org/10.1007/s11537-016-1619-9

Download citation

Keywords and phrases

  • integrable systems
  • Lenard–Magri scheme of integrability
  • rational pseudo-differential operators
  • symmetries

Mathematics Subject Classification (2010)

  • 37K10
  • 17B80
  • 35Q53
  • 35S05
  • 37K10