Abstract
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+/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.
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References
Ince, E.L.: Ordinary differential equations. New York: Dover Publications 1947
Ablowitz, M.J., Segur, H.: Phys. Rev. Lett.38, 1103–1106 (1977)
Airault, H.: Rational solutions of Painlevé equations (to appear)
Wu, T.T., McCoy, B.M., Tracy, C.A., Barouch, E.: Phys. Rev. B13, 316–371 (1976)
Barouch, E., McCoy, B.M., Wu, T.T.: Phys. Rev. Lett.31, 1409–1411 (1973)
McCoy, B.M., Tracy, C.A., Wu, T.T.: J. Math. Phys.18, 1058–1092 (1977)
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–XV
Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Commun. Pure Appl. Math.27, 97–133 (1976)
Fuchs, R.: Math. Ann.63, 301–321 (1906)
Ablowitz, M.J., Segur, H.: Stud. Appl. Math.57, 13–44 (1977)
Hastings, S.P., McLeod, J.B.: Univ. of Wisconsin, MRC Report No. 1861 (1978)
Ablowitz, M.J., Ramani, A., Segur, H.: Lett. Nuovo Cimento23, 333 (1978).
Two preprints: A connection between nonlinear evolution equations and ordinary differential equations ofP-type, I, II
Tracy, C.A.: Proc. NATO Advanced Study Institute on: Nonlinear equations in physics and mathematics, 1978, (ed. A. Barut). Dordrecht, Holland: Reidel 1978
Schlesinger, L.: J. Reine Angewandte Math.141, 96–145 (1912)
Garnier, R.: Ann. Ec. Norm. Sup.29, 1–126 (1912)
Birkhoff, G.D.: Trans. AMS10, 436–470 (1909)
Birkhoff, G.D.: Proc. Am. Acad. Arts Sci.49, 521–568 (1913)
Garnier, R.: Rend. Circ. Mat. Palermo,43, 155–191 (1919)
Davis, H.T.: Introduction to nonlinear differential and integral equations. New York: Dover Publications 1962
Choodnovsky, D.V., Choodnovsky, G.V.: Completely integrable class of mechanical systems connected with Korteweg-deVries and multicomponent Schrödinger equations. I. Preprint, École Polytechnique, 1978
Moser, J., Trubowitz, E.: (to appear)
Olver, F.W.J.: Asymptotics and special functions. New York: Academic Press 1974
Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: Stud. Appl. Math.53, 249–315 (1974)
Flaschka, H., Newell, A.C.: Springer Lecture Notes in Physics38, 355–440 (1975)
Airault, H., McKean, Jr., H.P., Moser, J.: Comm. Pure Appl. Math.30, 95–148 (1977)
Brieskorn, E.: Jber. Dt. Math.-Verein.78, 93–112 (1976)
Ueno, K.: Kyoto, RIMS master's thesis, Dec. 1978. RIMS Preprints 301, 302 (1979)
Sibuya, Y.: Proc. Int. Conf. Diff. Eq. pp. 709–738. (ed. H. A. Antosiewicz). New York: Academic Press 1975;
Bull. AMS83, 1075–1077 (1977)
Zakharov, V.E., Shabat, A.B.: Sov. Phys. JETP34, 62–69 (1972)
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)
Krichever, I.M.: Funkts. Anal. Prilozen11, 15–31 (1977)
Novikov, S.P.: Rocky Mt. J. Math.8, 83–94 (1978)
Newell, A.C.: Proc. Roy. Soc. London A365, 283–311 (1979)
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Communicated by A. Jaffe
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Flaschka, H., Newell, A.C. Monodromy- and spectrum-preserving deformations I. Commun.Math. Phys. 76, 65–116 (1980). https://doi.org/10.1007/BF01197110
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DOI: https://doi.org/10.1007/BF01197110