The Small Dispersion Limit of the Korteweg-De Vries Equation

  • Stephanos Venakides

Abstract

There are many physical systems which display shocks i.e. regions in space where the solution develops extremely large slopes. In general, such systems are too complicated to be treated by exact calculation and their properties are best studied through the proof of general theorems. A model of the formation and propagation of dispersive shocks in one space dimension, in which explicit calculation is possible, is given by the initial value problem for the Korteweg-de Vries equation:
$$ {u_{t}} - 6u{u_{x}} + {\varepsilon ^{2}}u{u_{{xxx}}} = 0 $$
(1.1a)
$$ u(x,o,\varepsilon ) = - v(x)$$
(1.1b)
in the limit ε → 0.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    V.S. Buslaev and V.N. Fomin, “An inverse scattering problem for the one-dimensional Schrödinger equation on the entire axis,” Vestnik Leningrad Univ. 17, 1962, 56–64 (In Russian).MathSciNetMATHGoogle Scholar
  2. [2]
    A. Cohen, T. Rappeler, “Scattering and Inverse Scattering for Steplike Potentials in the Schrödinger Equation,” Indiana U. Math J., Vol 34, #1, 1985, 127–180.MATHCrossRefGoogle Scholar
  3. [3]
    P. Deift, E. Trubowitz, “Inverse Scattering on the Line,” Comm. Pure Appl. Math. 32, 1979, 121–252.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    B.A. Dubrovin, V.B. Matveev, and S.P. Novikov, “Nonlinear equations of Korteweg-de Vries type, finite zoned linear operators, and Abelian varieties,” Uspekhi Mat. Nauk. 31, 1976, 55–136.MathSciNetMATHGoogle Scholar
  5. [5]
    F.J. Dyson, “Old and New Approaches to the Inverse Scattering Problem,” Studies in Math Physics, Princeton Series in Physics (Lieb, Simon, Wightman eds.) 1976.Google Scholar
  6. [6]
    L. Faddeev, “The Inverse Problem in the Quantum Theory of Scattering,” J. Math. Phys., Vol 4, #1, Jan. 1963, 72–104.MathSciNetADSCrossRefGoogle Scholar
  7. [7]
    H. Flaschka, M.G. Forest and D.W. McLaughlin, “Multiphase Averaging and the Inverse Spectral Solution of the Korteweg-de Vries Equation,” Comm. Pure Appl. Math. 33, 1980, 739–784.MathSciNetADSMATHCrossRefGoogle Scholar
  8. [8]
    C.S. Gardner, J.M. Green, M.D. Kruskal, R.M. Miura, “Method for solving the Korteweg-de Vries equation,” Phys. Rev. Lett. 19, (1967), 1095–1097.ADSMATHCrossRefGoogle Scholar
  9. [9]
    I.M. Gelfand, B.M. Levitan, “On the determination of a differential equation from its spectral function,” Izv, Akad. Nauk SSR, Ser. Math., 15 (309–60). Eng. Translation: Am. Math. Soc. Translation (2), 1, 253 (1955).MathSciNetGoogle Scholar
  10. [10]
    I. Kay, H.E. Moses, “Reflectionless Transmission through Dielectrics and Scattering Potentials,” J. Appl. Phys. 27, 1956. 1503–1508.ADSMATHCrossRefGoogle Scholar
  11. [11]
    P.D. Lax, “Integrals of Nonlinear Equations of Evolution and Solitary Waves,” Comm. Pure Appl. Math., Vol 21 (1968), 467–490.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    P.D. Lax and C.D. Levermore, “The Small Dispersion Limit of the Korteweg-de Vries Equation,” I, II, III Comm. Pure Appl. Math. 36, 1983, 253–290, 571–593, 809–829.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    H.P. McKean and E. Trubowitz, “Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points,” Comm. Pure Appl. Math. 29, 1976, 146–226.MathSciNetGoogle Scholar
  14. [14]
    H.P. McKean and P. vanMoerbeke, “The spectrum of Hill’s equation,” Invent. Math. 30, 1975, 217–274.MathSciNetADSMATHCrossRefGoogle Scholar
  15. [15]
    H.P. McKean, “Partial Differential Equations and Geometry,” Proc. Park City Conference, editor C.I. Byrnes, Marcel Dekker, Inc., New York, 1979, 237–252.Google Scholar
  16. [16]
    E. Trubowitz, “The inverse problem for periodic potentials,” Comm. Pure Appl. Math. 30, 1977, 321–337.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    S. Venakides, “The zero dispersion limit of the Korteweg-de Vries equation with non-trivial reflection coefficient,” Comm. Pure Appl. Math. 38, 1985, 125–155.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    S. Venakides, “The generation of modulated wavetrains in the solution of the Korteweg-de Vries equation,” Comm. Pure Appl. Math. 38, 1985, 883–909.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    S. Venakides, “The zero dispersion limit of the periodic kdV equation,” AMS Transactions in press.Google Scholar
  20. [20]
    G. Whitham, “Linear and Nonlinear Waves,” Wiley Interscience, New York, 1974.MATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • Stephanos Venakides
    • 1
  1. 1.Stanford UniversityUSA
  2. 2.MathematicsDuke UniversityDurhamUSA

Personalised recommendations