Resonances in Multifrequency Averaging Theory

  • S. Yu. Dobrokhotov
Part of the NATO ASI Series book series (NSSB, volume 320)

Abstract

The Whitham method allows to obtain rapidly oscillating asymptotic solutions of nonlinear equations with a small parameter ε which characterizes the dispersion. The principal term of such asymptotic solutions can be represented in the form
$$ u = f\left( {\frac{{S\left( {x,t} \right)}}{\varepsilon },x,t} \right)$$
(1.1)
, where f (τ, x, t) and S(x, t) are smooth functions and f is 2π-periodic in the argument τ. The function (1.1) describes the distribution of wave packets. Such solutions were used in different physical and mechanical problems (see, for examplc, the bibliography in [1-16); they satisfy the special Cauchy data
$$ u{{|}_{{t = 0}}} = f\left( {\frac{{{{S}^{0}}\left( x \right)}}{\varepsilon },x,0} \right)$$
(1.2)

Keywords

Manifold Soliton Eter Stimate Summing 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Whitham G.B., Linear and Nonlinear Waves, Wiley Interscience. New York. 1974.MATHGoogle Scholar
  2. 2.
    Scott A.C., Chu F.Y., McLaughlin D.W.The soliton: a new concept in applied science, Proc. IEEE 1 (1973), 1443.MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    Dobrokhotov S.Y., Maslov V.P., Finite-gap almost periodic solutions in WKB-approrimations, Itogi Nauki i Tehniki. Modern Problems of Mathematics 15 (1980). Moscow, VINITI AN SSSR, 3 - 94.Google Scholar
  4. 4.
    Dobrokhotov S.Y., Maslov V.P., Multiphase asiinptotics of nonlinear partial differential equations with a small parameter, Soy. Sci. Rev.- Math. Phys. Rev. 3 (1982), 221–311.MathSciNetMATHGoogle Scholar
  5. 5.
    Dubrovin B.A., Novikov S.P., Uspekhi Mat.Nauk XLIV (1989), 29.Google Scholar
  6. 6.
    Bikbaev R.F., KdV equation with finite-zone boundary conditions and Whithain’s deformations of Riemann surfaces, Func. anal. i priloj. 23:4 (1989), 1–10.MathSciNetGoogle Scholar
  7. 7.
    Krichever I.M., Spectral theory of two-dimensional periodic operators and ets applications, Uspekhi Mat. Nauk XLIV:2 (1989), 121–184.MathSciNetGoogle Scholar
  8. 8.
    Haberman R., The modulated phase shift for weakly dissipated non-linear oscillatory waves of the KdV type, Stud. in Appl. Math. LXXVIII:1 (1988), 73–90.MathSciNetGoogle Scholar
  9. 9.
    Maslov V.P., Asymptotic methods and perturbation theory, Nauka, Moscow, 1988.MATHGoogle Scholar
  10. 10.
    Flashka H., Forest M.G., McLaughlin D.W., Multiphase averaging and the inverse spectral solution of the KdV equation, Comm. Pure and Appl. Math. 33:6 (1980), 739–784.MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    Ablowitz M.A., Benny D.Y., The evolution of multiphase modes for non-linear dispersive waves, Stud. Appl. Math. 49:3 (1970).Google Scholar
  12. 12.
    Zakharov V.E., Manakov S.V., Novikov S.P., Pitaevsky L.P., The soliton theory, Nauka, Moscow, 1979.Google Scholar
  13. 13.
    Dubrovin B.A., Matveev V.B., Novikov S.P., Non-linear KdV-equations, finite-gap linear operators and Abelian manifolds, Uspekhi Mat. Nauk 31:1 (1976), 55–136.MathSciNetMATHGoogle Scholar
  14. 14.
    Luke J.C., A perturbation method for non-linear dispersive wave problems, Proc. Roy. Soc. A292:1430 (1966), 403–412.MathSciNetADSGoogle Scholar
  15. 15.
    Dobrokhotov S.Yu., Maslov V.P., Finite-gap almost periodic solutions in asymptotic expansions, Math. Studies 47 (1981), Noth Holland Publ. Comp., Amsterdam, 1–27.MathSciNetGoogle Scholar
  16. 16.
    Dobrokhotov S.Yu., Maslov V.P., Solution asymptotics of mixed problem for the non-linear wave equation h 2u + α sinh u = 0, Uspekhi Mat. Nauk 34:3 (1979), 225–226.MathSciNetGoogle Scholar
  17. 17.
    Dobrokhotov S.Yu., Maslov V.P., Boundary reflection problem for the equation h 2u+ + α sinh u =❚ 0 and finite-gap quasiperiodic solutions, Func. Anal. i Priloj. 13:3 (1973), 79–80.MathSciNetGoogle Scholar
  18. 18.
    Dobrokhotov S.Yu., Vorob’cv Y.M., Basic systems on the torus generated by finite-zone integration of the KdV quation, Matem. Zametki 47:1 (1990), 47–61.Google Scholar
  19. 19.
    Dobrokhotov S.Yu., Resonance correction for adiabatically perturbed finite-zone almost-periodic solution of KdV equation, Matem. Zametki 44:4 (1988), 551–554.Google Scholar
  20. 20.
    Bensousan A., Lions. J.L., Papanicolaou G., Asymptotic analysis for periodic structures, Nothlloll. Publ. Comp., Amsterdam, 1978.Google Scholar
  21. 21.
    Dobrokhotov S.Yu., Resonances in asymptotic solutions of Cauchy problem for the Schrödinger equation with rapidly oscillating finite-zone potential, Matem. Zametki 44:3 (1988), 319–340.MathSciNetGoogle Scholar
  22. 22.
    Dobrokhotov S.Yu., Vorob’ev Y.M., Completness of the system of eigenfunctions of a none!liptic operator on the torus generated by the Hill operator with finite-zone potential, Func. Anal. i Priloj. 22:2 (1988), 65–66.MathSciNetMATHGoogle Scholar
  23. 23.
    Dobrokhotov S.Yu., Krichever I.M., Multiphase solutions of Benjamin-Ono equation and its averaging, Matem. Zametki 49:6 (1991).Google Scholar
  24. 24.
    Krichever I.M., Hessians of integrals for KdV-equations and perturbations of finite-zone solutions, Dokl. AN SSSR 270:6 (1983), 1312–1317.MathSciNetGoogle Scholar
  25. 25.
    Kucherenko V.V., Asymptotic solution of system \( A\left( {x, - ih\left( {\partial /\partial x} \right)} \right)u = 0 \) case of characteristic of varying multiplicity, Izv. AN SSSR, ser. Math. 38:3 (1974), 378–383.Google Scholar
  26. 26.
    Dobrokhotov S.Yu., Doctor thesis (1989), LOMI AN SSSR, Leningrad.Google Scholar
  27. 27.
    Kaup D.J., A perturbation theory for inverse scattering transforms, SIAM J. Appl. Math. 31:1 (1976), 121–132.MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Maslov V.P., Operational methods, MIR, Moscow, 1976.MATHGoogle Scholar
  29. 29.
    Babich V.M., Buldyrev V.S., Asymptotic method in short wave problems, Nauka, Moscow, 1972.Google Scholar
  30. 30.
    Lazutkin V.E., Convex billiard and eigenfunctions of Laplace operator, L.GU, Leningrad, 1981.MATHGoogle Scholar
  31. 31.
    Krakhnov A.D., On asymptotics of eigenvalues of pseudodifferential operators and invariant tori, Uspekhi Mat.Nauk XXXI:3 (1976), 217–218.Google Scholar

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • S. Yu. Dobrokhotov

There are no affiliations available

Personalised recommendations