Remote Access Transactions of the Moscow Mathematical Society

Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)

Request Permissions   Purchase Content 


On holomorphic solutions of equations of Korteweg-de Vries type

Author: A. V. Domrin
Translated by: V. E. Nazaikinskii
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva.
Journal: Trans. Moscow Math. Soc. 2012, 193-206
MSC (2010): Primary 35Q53; Secondary 30B40
Published electronically: March 21, 2013
MathSciNet review: 3184975
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that, for any of the equations indicated in the title, every solution locally holomorphic in $ x$ and $ t$ admits global meromorphic continuation in $ x$ for each $ t$ with trivial monodromy at each pole. By way of application, we describe all possible envelops of meromorphy of local holomorphic solutions of the Boussinesq equation.

References [Enhancements On Off] (What's this?)

  • 1. V. G. Drinfel′d and V. V. Sokolov, Lie algebras and equations of Korteweg-de Vries type, Current problems in mathematics, Vol. 24, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 81–180 (Russian). MR 760998
  • 2. Graeme Segal and George Wilson, Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. 61 (1985), 5–65. MR 783348
  • 3. B. V. Shabat, Introduction to complex analysis. Part II, Translations of Mathematical Monographs, vol. 110, American Mathematical Society, Providence, RI, 1992. Functions of several variables; Translated from the third (1985) Russian edition by J. S. Joel. MR 1192135
  • 4. A. V. Domrin, Meromorphic extension of solutions of soliton equations, Izv. Ross. Akad. Nauk Ser. Mat. 74 (2010), no. 3, 23–44 (Russian, with Russian summary); English transl., Izv. Math. 74 (2010), no. 3, 461–480. MR 2682370, 10.1070/IM2010v074n03ABEH002494
  • 5. I. M. Kričever, Algebraic curves and commuting matrix differential operators, Funkcional. Anal. i Priložen. 10 (1976), no. 2, 75–76 (Russian). MR 0413179
  • 6. I. M. Gel′fand and L. A. Dikiĭ, Fractional powers of operators, and Hamiltonian systems, Funkcional. Anal. i Priložen. 10 (1976), no. 4, 13–29 (Russian). MR 0433508
  • 7. L. A. Dickey, Soliton equations and Hamiltonian systems, 2nd ed., Advanced Series in Mathematical Physics, vol. 26, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. MR 1964513
  • 8. B. A. Dubrovin, Igor Moiseevich Krichever, and S. P. Novikov, Integrable systems. I, Current problems in mathematics. Fundamental directions, Vol. 4, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985, pp. 179–284, 291 (Russian). MR 842910
  • 9. Richard Beals, Percy Deift, and Carlos Tomei, Direct and inverse scattering on the line, Mathematical Surveys and Monographs, vol. 28, American Mathematical Society, Providence, RI, 1988. MR 954382
  • 10. Ju. I. Manin, Algebraic aspects of nonlinear differential equations, Current problems in mathematics, Vol. 11 (Russian), Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii, Moscow, 1978, pp. 5–152. (errata insert) (Russian). MR 0501136
  • 11. F. Gesztesy, D. Race, K. Unterkofler, and R. Weikard, On Gel′fand-Dickey and Drinfel′d-Sokolov systems, Rev. Math. Phys. 6 (1994), no. 2, 227–276. MR 1269299, 10.1142/S0129055X94000122
  • 12. I. M. Kričever, Integration of nonlinear equations by the methods of algebraic geometry, Funkcional. Anal. i Priložen. 11 (1977), no. 1, 15–31, 96 (Russian). MR 0494262
  • 13. S. P. Novikov, A periodic problem for the Korteweg-de Vries equation. I, Funkcional. Anal. i Priložen. 8 (1974), no. 3, 54–66 (Russian). MR 0382878
  • 14. D. H. Sattinger and J. S. Szmigielski, Factorization and the dressing method for the Gel′fand-Dikiĭ hierarchy, Phys. D 64 (1993), no. 1-3, 1–34. MR 1214545, 10.1016/0167-2789(93)90247-X
  • 15. A. V. Domrin, The Riemann problem and matrix-valued potentials with a converging Baker-Akhiezer function, Teoret. Mat. Fiz. 144 (2005), no. 3, 453–471 (Russian, with Russian summary); English transl., Theoret. and Math. Phys. 144 (2005), no. 3, 1264–1278. MR 2191841, 10.1007/s11232-005-0158-y
  • 16. R. Weikard, On commuting differential operators, Electron. J. Differential Equations (2000), No. 19, 11 pp. (electronic). MR 1744086
  • 17. E. L. Ince, Ordinary differential equations, Longmans, Green & Co, London, 1927.
  • 18. A. P. Veselov and A. B. Shabat, A dressing chain and the spectral theory of the Schrödinger operator, Funktsional. Anal. i Prilozhen. 27 (1993), no. 2, 1–21, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 27 (1993), no. 2, 81–96. MR 1251164, 10.1007/BF01085979
  • 19. G. Pólya and G. Szegő, Problems and theorems in analysis. Vol. II, Revised and enlarged translation by C. E. Billigheimer of the fourth German edition, Springer-Verlag, New York-Heidelberg, 1976. Theory of functions, zeros, polynomials, determinants, number theory, geometry; Die Grundlehren der Mathematischen Wissenschaften, Band 216. MR 0396134
  • 20. A. V. Domrin, Remarks on a local version of the method of the inverse scattering problem, Tr. Mat. Inst. Steklova 253 (2006), no. Kompleks. Anal. i Prilozh., 46–60 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 2 (253) (2006), 37–50. MR 2338686
  • 21. A. V. Domrin, The local holomorphic Cauchy problem for soliton equations of parabolic type, Dokl. Akad. Nauk 420 (2008), no. 1, 14–17 (Russian); English transl., Dokl. Math. 77 (2008), no. 3, 332–335. MR 2462096, 10.1134/S1064562408030034
  • 22. A. A. Bolibrukh, Inverse monodromy problems in analytic theory of differential equations, MTsMNO, Moscow, 2009. (Russian)
  • 23. G. S. Salehov and V. R. Fridlender, On a problem inverse to the Cauchy-Kovalevskaya problem, Uspehi Matem. Nauk (N.S.) 7 (1952), no. 5(51), 169–192 (Russian). MR 0057430
  • 24. Otto Forster, Riemannsche Flächen, Springer-Verlag, Berlin-New York, 1977 (German). Heidelberger Taschenbücher, Band 184. MR 0447557
  • 25. L. A. Takhtadzhyan and L. D. Faddeev, Gamiltonov podkhod v teorii solitonov, “Nauka”, Moscow, 1986 (Russian). MR 889051

Similar Articles

Retrieve articles in Transactions of the Moscow Mathematical Society with MSC (2010): 35Q53, 30B40

Retrieve articles in all journals with MSC (2010): 35Q53, 30B40

Additional Information

A. V. Domrin
Affiliation: Faculty of Mechanics and Mathematics, Moscow State University, 1 Leninskie Gory, 119991 Moscow, Russian Federation

Keywords: Soliton equation, holomorphic solution, analytic continuation
Received by editor(s): July 28, 2012
Published electronically: March 21, 2013
Additional Notes: Supported by RFBR grants nos. 11-01-12033-ofi-m, 11-01-00495-a-2011, and 10-01-00178-a
Article copyright: © Copyright 2013 American Mathematical Society