Advertisement

Theoretical and Mathematical Physics

, Volume 171, Issue 1, pp 435–441 | Cite as

Differential operators commuting in the principal part

  • R. A. Gabiev
  • A. B. Shabat
Article

Abstact

We consider commuting differential operators with two independent variables of general form and obtain general necessary commutativity conditions for low-order operators. We show that these conditions allow classifying commuting pairs of operators whose coefficients are linear functions of the independent variables.

Keywords

commutative ring of differential operators Schur’s formula classification 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    I. Schur, Sitzungsber. Berl. Math. Gen., 4, 2–8 (1905).Google Scholar
  2. 2.
    A. V. Mikhailov, A. B. Shabat, and R. I. Yamilov, Russ. Math. Surveys, 42, 1–63 (1987).MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    J. Hietarinta, Phys. Lett. A, 246, 97–104 (1998).MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    V. G. Marikhin, Theor. Math. Phys., 168, 1041–1047 (2011).CrossRefGoogle Scholar
  5. 5.
    A. B. Shabat, Lectures on Soliton Theory, Karachai-Cherkess State Univ., Karachaevsk (2008).Google Scholar
  6. 6.
    P. Appell, Sur les fonctions hypergéométriques de plusieurs variables, les polynomes d’Hermite et autres fonctions sphériques dans l’hyperespace, Gauthier-Villars, Paris (1925).zbMATHGoogle Scholar
  7. 7.
    A. Gonzalez-Lopez, N. Kamran, and P. Olver, J. Phys. A, 24, 3995–4008 (1991).MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.Aliev Karachai-Cherkess State UniversityKarachaevskRussia
  2. 2.Landau Institute for Theoretical PhysicsChernogolovka, Moscow OblastRussia

Personalised recommendations