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Nonisospectral Symmetries of the KdV Equation and the Corresponding Symmetries of the Whitham Equations

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Singular Limits of Dispersive Waves

Part of the book series: NATO ASI Series ((NSSB,volume 320))

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Abstract

In our paper we construct a new infinite family of symmetries of the Whitham equations (averaged Korteveg-de-Vries equation). In contrast with the ordinary hydrodynamic-type flows these symmetries are nonhomogeneous (i.e. they act nontrivially at the constant solutions), are nonlocal, explicitly depend upon space and time coordinates and form a noncommutative algebra, isomorphic to the algebra of the polynomial vector fields in the plain.

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References

  1. G.B. Whitham. Linear and nonlinear waves. N.Y., Wiley, (1974).

    MATH  Google Scholar 

  2. H. Flaschka, G. Forest, D.W. MkLaughlin. Commun. Pure Appl. Math. 33, no.6 (1980).

    Google Scholar 

  3. Peter D.Lax, C.David Levermore. Commun. Pure Appl. Math., 36, pp.253–290, 571–593, 809–830 (1983).

    Google Scholar 

  4. B.A. Dubrovin, S.P. Novikov. Sov. Math. Doklady 27, p.665 (1983); Russian Math. Surveys 44:6 (1989), p.35.

    Google Scholar 

  5. S.P. Tzarev. Math. USSR Izvestiya, 54, no. 5 (1990).

    Google Scholar 

  6. B.A. Dubrovin. Hamiltonian formalism of Whitham-type hierarchies and topological Landau-Ginzburg models. Preprint 1991.

    Google Scholar 

  7. S.P. Tzarev. Soviet. Math. Dokl. 31, pp.488–491 (1985).

    Google Scholar 

  8. I.M. Krichever. Funct. Anal. Appl. 11 (1988), p.15.

    MathSciNet  Google Scholar 

  9. F. Calogero, A. Degasperis. Lett. Nuov. Cim. 22, p.420 (1978).

    Article  MathSciNet  ADS  Google Scholar 

  10. V.A. Belinskii, V.E. Zakharov. JETP 12, p.1953 (1978).

    Google Scholar 

  11. Maison. Phys. Rev. Lett. 41, p.521 (1978).

    Article  MathSciNet  ADS  Google Scholar 

  12. G. Baruccy, T. Redge J. Math. Phys. 18 No 6, p.1149 (1976).

    Article  ADS  Google Scholar 

  13. N.H. Ibragimov, A.B. Shabat. Dokl. Akad. Nauk SSSR 244 (1979), p.1.

    MathSciNet  Google Scholar 

  14. H.H. Chen, Y.C. Lee, J.E. Lin. Physica 9D, 9, No 3, pp.439–445 (1983).

    MathSciNet  ADS  MATH  Google Scholar 

  15. F.J. Schwarz. J. Phys. Soc. Japan, 51 No 8, p. 2387 (1982).

    Article  MathSciNet  ADS  Google Scholar 

  16. B. Fuckssteiner. Progr. Thoer. Phys., 70, pp. 1508–1522 (1983); W. Oevel, B.Fuckssteiner Phys. Lett. 88A, p.323 (1982).

    Google Scholar 

  17. A.Yu. Orlov, E.I. Schulman. Additional Symmetries of the Integrable Systems and Conformal Algebra Representations. Preprint IA and E, No 217, Novosibirsk (1984); A.Yu. Orlov, E.I. Schulman. Theor. Math. Phys., 64, pp. 323–327 (1985) (Russian); A.Yu. Orlov, E.I. Schulman. Additional Symmetries for 2-D Integrable Systems. Preprint IA and E, No 277, Novosibirsk (1985); A.Yu. Orlov, E.I. Schulman. Lett. Math. Phys. 12, pp. 171–179 (1986). A.Yu. Orlov in Proceedings Int. Workshop “Plasma theory and …” in Kiev 1987, World Scientific (1988) p.116–134.

    Google Scholar 

  18. A.S. Fokas, P.M. Santini. Commun. Math. Phys. 116, p.449 (1088).

    Article  MathSciNet  ADS  Google Scholar 

  19. I.M. Krichever, S.P. Novikov. Funct. Anal. i Prilog. 21, No 2, pp. 46–63 (1987); 21, No 4, pp. 47–61 (1987); 23, No 1, pp. 41–56 (1989).

    Google Scholar 

  20. P.G. Grinevich, A.Yu. Orlov. Virasoro action on Riemann surfaces, Grassmannians, det ā j and Segal-Vilson т function. in “Probimes of modern quantum field theory”, ed. A.A. Belavin, A.U. Klimyk, A.B. Zamolodchikov, Springer 1989.

    Google Scholar 

  21. P.G. Grinevich, A.Yu. Orlov. Funct. Anal. Appl. 24 no. 1, (1990); Higher symmetries of KP equation and the Virasoro action on Riemann surfaces. in “Nonlinear evolution equations and dynamical systems”, ed. S. Carillo, O. Ragnisco, Springer 1990.

    Google Scholar 

  22. A. Gerasimov, A. Marshakov, A. Mironov, A. Morozov, A. Orlov. Nuclear Phys. B 357, pp. 565–618 (1991).

    Article  MathSciNet  ADS  Google Scholar 

  23. I.M. Krichever. “The dispersionless Lax equations and topological minimal models”. Preprint 1991 Torino; Lecture at the Sakharov memory congress (1991).

    Google Scholar 

  24. G. Springer. Introduction to Riemann surfaces. Addison - Wesley Publishing Company, Inc. Reading, Massachusetts, USA (1957).

    MATH  Google Scholar 

  25. M. Shiffer, D.K. Spencer. Functionals of finite Riemann surfaces. Princeton, New Jersey (1953).

    Google Scholar 

  26. D. Mumford. Tata lectures on Theta. Birkhäuser, Boston - Basel - Stuttgart (1983).

    Google Scholar 

  27. R. Dijkgraaf, H. Verlinder, E. Verlinder. “Topological strings in d < 1”. Princeton preprint PUPT - 1024, IASSNS-HEP - 90/71.

    Google Scholar 

  28. V.E. Zakharov, S.V. Manakov, S.P. Novikov, L.P. Pitaevsky. Soliton theory. Plenum, New York, 1984.

    Google Scholar 

  29. I.M. Krichever. Docl. Akad. Nauk. SSSR, 227, No 2, pp. 291–294 (1977).

    Google Scholar 

  30. I.M. Krichever. Uspekhy. Mat. Nauk. 44, No 2, pp. 121–184 (1989).

    MathSciNet  Google Scholar 

  31. I.M. Gel’fand, L.A. Dikii. Funct. Anal. Pril. 10, No 4, pp. 13–29 (1976).

    MathSciNet  MATH  Google Scholar 

  32. H. McKean, E. Trubowitz. Commun Pure. Appl. Math. 29, pp. 143–226 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  33. B.A. Dubrovin. Uspekhi. Math. Nauk. 36, No 2, pp. 11–40 (1981).

    MathSciNet  Google Scholar 

  34. N. Kawamoto, Yu. Namikava, A. Tsuchiya, Ya. Yamada. Commun. Math. Phys. 116, No. 2, pp.247–308 (1988).

    Article  ADS  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. L.D.Landau Institute for Theoretical Physics, Kosygina 2, 117940, Moscow, USSR

    P. G. Grinevich

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  1. P. G. Grinevich
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Editor information

Editors and Affiliations

  1. University of Arizona, Tucson, Arizona, USA

    N. M. Ercolani  & C. D. Levermore  & 

  2. Landau Institute for Theoretical Physics, Moscow, Russia

    I. R. Gabitov

  3. Ecole Normale Superieure de Lyon, Lyon, France

    D. Serre

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Grinevich, P.G. (1994). Nonisospectral Symmetries of the KdV Equation and the Corresponding Symmetries of the Whitham Equations. In: Ercolani, N.M., Gabitov, I.R., Levermore, C.D., Serre, D. (eds) Singular Limits of Dispersive Waves. NATO ASI Series, vol 320. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-2474-8_6

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  • DOI: https://doi.org/10.1007/978-1-4615-2474-8_6

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-6054-4

  • Online ISBN: 978-1-4615-2474-8

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