Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 044, 11 pages      arXiv:1105.5774      https://doi.org/10.3842/SIGMA.2012.044

Commuting Differential Operators of Rank 3 Associated to a Curve of Genus 2

Dafeng Zuo a, b
a) School of Mathematical Science, University of Science and Technology of China, Hefei 230026, P.R. China
b) Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, P.R. China

Received March 12, 2012, in final form July 12, 2012; Published online July 15, 2012

Abstract
In this paper, we construct some examples of commuting differential operators L1 and L2 with rational coefficients of rank 3 corresponding to a curve of genus 2.

Key words: commuting differential operators; rank 3; genus 2.

pdf (334 kb)   tex (16 kb)

References

  1. Burchnall J.L., Chaundy T.W., Commutative ordinary differential operators, Proc. Lond. Math. Soc. s2-21 (1923), 420-440.
  2. Burchnall J.L., Chaundy T.W., Commutative ordinary differential operators, Proc. R. Soc. Lond. Ser. A 118 (1928), 557-583.
  3. Burchnall J.L., Chaundy T.W., Commutative ordinary differential operators. II. The identity Pn=Qm, Proc. R. Soc. Lond. Ser. A 134 (1931), 471-485.
  4. Dixmier J., Sur les algèbres de Weyl, Bull. Soc. Math. France 96 (1968), 209-242.
  5. Dubrovin B.A., Periodic problems for the Korteweg-de Vries equation in the class of finite band potentials, Funct. Anal. Appl. 9 (1975), 215-223.
  6. Dubrovin B.A., Matveev V.B., Novikov S.P., Non-linear equations of Korteweg-de Vries type, finite-zone linear operators, and Abelian varieties, Russ. Math. Surv. 31 (1976), no. 1, 59-146.
  7. Grinevich P.G., Rational solutions for the equation of commutation of differential operators, Funct. Anal. Appl. 16 (1982), 15-19.
  8. Kasman A., Darboux transformations from n-KdV to KP, Acta Appl. Math. 49 (1997), 179-197.
  9. Kasman A., Rothstein M., Bispectral Darboux transformations: the generalized Airy case, Phys. D 102 (1997), 159-176, q-alg/9606018.
  10. Krichever I.M., Commutative rings of ordinary linear differential operators, Funct. Anal. Appl. 12 (1978), 175-185.
  11. Krichever I.M., Integration of nonlinear equations by the methods of algebraic geometry, Funct. Anal. Appl. 11 (1977), 12-26.
  12. Krichever I.M., Methods of algebraic geometry in the theory of non-linear equations, Russ. Math. Surv. 32 (1977), no. 6, 185-213.
  13. Krichever I.M., Novikov S.P., Holomorphic bundles and nonlinear equations. Finite-gap solutions of rank 2, Dokl. Akad. Nauk SSSR 247 (1979), 33-37.
  14. Krichever I.M., Novikov S.P., Holomorphic bundles over Riemann surfaces and the Kadomtsev-Petviashvili equation. I, Funct. Anal. Appl. 12 (1978), 276-286.
  15. Latham G.A., Previato E., Darboux transformations for higher-rank Kadomtsev-Petviashvili and Krichever-Novikov equations, Acta Appl. Math. 39 (1995), 405-433.
  16. Latham G., Previato E., Higher rank Darboux transformations, in Singular Limits of Dispersive Waves (Lyon, 1991), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 320, Plenum, New York, 1994, 117-134.
  17. Mironov A.E., A ring of commuting differential operators of rank 2 corresponding to a curve of genus 2, Sb. Math. 195 (2004), 711-722.
  18. Mironov A.E., Commuting rank 2 differential operators corresponding to a curve of genus 2, Funct. Anal. Appl. 39 (2005), 240-243.
  19. Mironov A.E., On commuting differential operators of rank 2, Sib. Èlektron. Mat. Izv. 6 (2009), 533-536.
  20. Mironov A.E., Self-adjoint commuting differential operators and commutative subalgebras of the Weyl algebra, arXiv:1107.3356.
  21. Mokhov O.I., Commuting differential operators of rank 3, and nonlinear equations, Math. USSR Izv. 35 (1990), 629-655.
  22. Mokhov O., On commutative subalgebras of the Weyl algebra that are related to commuting operators of arbitrary rank and genus, arXiv:1201.5979.
  23. Novikov S.P., The periodic problem for the Korteweg-de vries equation, Funct. Anal. Appl. 8 (1974), 236-246.
  24. Novikov S.P., Grinevich P.G., Spectral theory of commuting operators of rank two with periodic coefficients, Funct. Anal. Appl. 16 (1982), 19-21.
  25. Previato E., Seventy years of spectral curves: 1923-1993, in Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 419-481.
  26. Previato E., Wilson G., Differential operators and rank 2 bundles over elliptic curves, Compositio Math. 81 (1992), 107-119.
  27. Previato E., Wilson G., Vector bundles over curves and solutions of the KP equations, in Theta Functions - Bowdoin 1987, Part 1 (Brunswick, ME, 1987), Proc. Sympos. Pure Math., Vol. 49, Amer. Math. Soc., Providence, RI, 1989, 553-569.
  28. Shabat A.B., Elkanova Z.S., Commuting differential operators, Theoret. and Math. Phys. 162 (2010), 276-285.
  29. Wilson G., Bispectral commutative ordinary differential operators, J. Reine Angew. Math. 442 (1993), 177-204.

Previous article  Next article   Contents of Volume 8 (2012)