Communications in Mathematical Physics

, Volume 76, Issue 1, pp 65–116 | Cite as

Monodromy- and spectrum-preserving deformations I

  • Hermann Flaschka
  • Alan C. Newell


A method for solving certain nonlinear ordinary and partial differential equations is developed. The central idea is to study monodromy preserving deformations of linear ordinary differential equations with regular and irregular singular points. The connections with isospectral deformations and with classical and recent work on monodromy preserving deformations are discussed. Specific new results include the reduction of the general initial value problem for the Painlevé equations of the second type and a special case of the third type to a system of linear singular integral equations. Several classes of solutions are discussed, and in particular the general expression for rational solutions for the second Painlevé equation family is shown to be −d/dx ln(Δ+), where Δ+ and Δ are determinants. We also demonstrate that each of these equations is an exactly integrable Hamiltonian system. The basic ideas presented here are applicable to a broad class of ordinary and partial differential equations; additional results will be presented in a sequence of future papers.


Neural Network Integral Equation Partial Differential Equation Ordinary Differential Equation Singular Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Ince, E.L.: Ordinary differential equations. New York: Dover Publications 1947Google Scholar
  2. 2.
    Ablowitz, M.J., Segur, H.: Phys. Rev. Lett.38, 1103–1106 (1977)Google Scholar
  3. 3.
    Airault, H.: Rational solutions of Painlevé equations (to appear)Google Scholar
  4. 4.
    Wu, T.T., McCoy, B.M., Tracy, C.A., Barouch, E.: Phys. Rev. B13, 316–371 (1976)Google Scholar
  5. 5.
    Barouch, E., McCoy, B.M., Wu, T.T.: Phys. Rev. Lett.31, 1409–1411 (1973)Google Scholar
  6. 6.
    McCoy, B.M., Tracy, C.A., Wu, T.T.: J. Math. Phys.18, 1058–1092 (1977)Google Scholar
  7. 7.
    Satō, M., Miwa, T., Jimbo, M.: A series of papers entitled Holonomic Quantum Fields: I. Publ. RIMS, Kyoto Univ.14, 223–267 (1977); II. Publ. RIMS, Kyoto Univ.15, 201–278 (1979); III. Publ. RIMS Kyoto Univ.15, 577–629 (1979). IV., V. RIMS Preprints 263 (1978), and 267 (1978). The paper we refer to most often is III. See also a series of short notes: Studies on holonomic quantum fields, I–XVGoogle Scholar
  8. 8.
    Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Commun. Pure Appl. Math.27, 97–133 (1976)Google Scholar
  9. 9.
    Fuchs, R.: Math. Ann.63, 301–321 (1906)Google Scholar
  10. 10.
    Ablowitz, M.J., Segur, H.: Stud. Appl. Math.57, 13–44 (1977)Google Scholar
  11. 11.
    Hastings, S.P., McLeod, J.B.: Univ. of Wisconsin, MRC Report No. 1861 (1978)Google Scholar
  12. 12.(i)
    Ablowitz, M.J., Ramani, A., Segur, H.: Lett. Nuovo Cimento23, 333 (1978).Google Scholar
  13. 12.(ii)
    Two preprints: A connection between nonlinear evolution equations and ordinary differential equations ofP-type, I, IIGoogle Scholar
  14. 13.
    Tracy, C.A.: Proc. NATO Advanced Study Institute on: Nonlinear equations in physics and mathematics, 1978, (ed. A. Barut). Dordrecht, Holland: Reidel 1978Google Scholar
  15. 14.
    Schlesinger, L.: J. Reine Angewandte Math.141, 96–145 (1912)Google Scholar
  16. 15.
    Garnier, R.: Ann. Ec. Norm. Sup.29, 1–126 (1912)Google Scholar
  17. 16.
    Birkhoff, G.D.: Trans. AMS10, 436–470 (1909)Google Scholar
  18. 17.
    Birkhoff, G.D.: Proc. Am. Acad. Arts Sci.49, 521–568 (1913)Google Scholar
  19. 18.
    Garnier, R.: Rend. Circ. Mat. Palermo,43, 155–191 (1919)Google Scholar
  20. 19.
    Davis, H.T.: Introduction to nonlinear differential and integral equations. New York: Dover Publications 1962Google Scholar
  21. 20.
    Choodnovsky, D.V., Choodnovsky, G.V.: Completely integrable class of mechanical systems connected with Korteweg-deVries and multicomponent Schrödinger equations. I. Preprint, École Polytechnique, 1978Google Scholar
  22. 21.
    Moser, J., Trubowitz, E.: (to appear)Google Scholar
  23. 22.
    Olver, F.W.J.: Asymptotics and special functions. New York: Academic Press 1974Google Scholar
  24. 23.
    Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: Stud. Appl. Math.53, 249–315 (1974)Google Scholar
  25. 24.
    Flaschka, H., Newell, A.C.: Springer Lecture Notes in Physics38, 355–440 (1975)Google Scholar
  26. 25.
    Airault, H., McKean, Jr., H.P., Moser, J.: Comm. Pure Appl. Math.30, 95–148 (1977)Google Scholar
  27. 26.
    Brieskorn, E.: Jber. Dt. Math.-Verein.78, 93–112 (1976)Google Scholar
  28. 27.
    Ueno, K.: Kyoto, RIMS master's thesis, Dec. 1978. RIMS Preprints 301, 302 (1979)Google Scholar
  29. 28.(i)
    Sibuya, Y.: Proc. Int. Conf. Diff. Eq. pp. 709–738. (ed. H. A. Antosiewicz). New York: Academic Press 1975;Google Scholar
  30. 28.(ii)
    Bull. AMS83, 1075–1077 (1977)Google Scholar
  31. 29.
    Zakharov, V.E., Shabat, A.B.: Sov. Phys. JETP34, 62–69 (1972)Google Scholar
  32. 30.
    Zakharov, V.E.: Paper at I. G. Petrovskii Memorial Converence, Moscow State Univ., Jan. 1976 (this paper has been referred to in many subsequent publications, but has apparently never been published)Google Scholar
  33. 31.
    Krichever, I.M.: Funkts. Anal. Prilozen11, 15–31 (1977)Google Scholar
  34. 32.
    Novikov, S.P.: Rocky Mt. J. Math.8, 83–94 (1978)Google Scholar
  35. 33.
    Newell, A.C.: Proc. Roy. Soc. London A365, 283–311 (1979)Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Hermann Flaschka
    • 1
  • Alan C. Newell
    • 1
  1. 1.Department of Mathematics and Computer ScienceClarkson College of TechnologyPotsdamUSA

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