On solitary water-waves of finite amplitude

  • C. J. Amick
  • J. F. Toland
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References

  1. 1.
    Bary, N. K., A treatise on trigonometric series, Vols. I & II. Pergamon, 1964.Google Scholar
  2. 2.
    Beale, J. T., The existence of solitary water waves. Comm. Pure Appl. Math. 30 (1977), 373–389.Google Scholar
  3. 3.
    Benjamin, T. B., Internal waves of finite amplitude and permanent form. J. Fluid Mech., 25 (2) (1966), 241–270.Google Scholar
  4. 4.
    Benjamin, T. B., Internal waves of permanent form in fluids of great depth. J. Fluid Mech., 29 (3) (1967), 559–592.Google Scholar
  5. 5.
    Benjamin, T. B., An exact theory of finite steady waves in continuously stratified fluids. University of Essex, Fluid Mechanics Research Institute Report No. 48 (1973).Google Scholar
  6. 6.
    Benjamin, T. B., Lectures on Nonlinear Waves. Nonlinear Wave Motion Proc. Summer Sem. Potsdam (N.Y.) 1972. Lectures in Applied Math. (15), American Math. Soc., Providence, R.I. (1974), 3–47.Google Scholar
  7. 7.
    Bona, J. L., & D. K. Bose, Fixed point theorems for Fréchet spaces and the existence of solitary waves. Nonlinear Wave Motion Proc. Summer Sem. Potsdam (N.Y.) 1972. Lectures in Applied Math. (15), American Math. Soc., Providence, R.I. (1974), 175–177.Google Scholar
  8. 8.
    Bona, J. L., & D. K. Bose, Solitary wave solutions for unidirectional wave equations having general form of nonlinearity and dispersion. University of Essex, Fluid Mechanics Research Institute Report No. 99, to appear in Proc. 1st American-Romanian Conf. on Operator Theory, Iasi, March, 1977.Google Scholar
  9. 9.
    Bona, J. L., Bose, D. K., & T. B. Benjamin, Solitary wave solutions for some model equations for waves in nonlinear dispersive media. Appl. Methods Funct. Anal. Probl. Mech., IUTAM/IMU Symp. Marseilles, 1975. Lecture Notes in Mathematics No. 503. Springer-Verlag, (1976), 207–218.Google Scholar
  10. 10.
    Bona, J. L., Bose, D. K., & R. E. L. Turner, Finite amplitude steady waves in stratified fluids, in preparation.Google Scholar
  11. 11.
    Boussinesq, J., Théorie de l'intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire. Comptes Rendus, June 19, (1871).Google Scholar
  12. 12.
    Byatt-Smith, J. G. B., & M. S. Longuet-Higgins, On the speed and profile of steep solitary waves. Proc. R. Soc. Lond. A, 350 (1976), 175–189.Google Scholar
  13. 13.
    Cokelet, E. D., Steep gravity waves in water of arbitrary uniform depth. Phil. Trans. R. Soc. Lond., 286 (1335) (1977), 183–230.Google Scholar
  14. 14.
    Dancer, E. N., Global solution branches for positive mappings. Arch. Rational Mech. Anal., 52 (1973), 181–192.Google Scholar
  15. 15.
    Davis, R. E., & A. Acrivos, Solitary internal waves in deep water. J. Fluid Mech., 29 (3) (1967), 593–607.Google Scholar
  16. 16.
    Friedrichs, K. O., & D. H. Hyers, The existence of solitary waves. Comm. Pure Appl. Math., 7 (1954), 517–550.Google Scholar
  17. 17.
    Günter, N. M., Potential theory and its application to basic problems of mathematical physics. Ungar, 1964.Google Scholar
  18. 18.
    Hansen, E. R., A table of series and products. Prentice-Hall, Englewood Cliffs, N.J., 1975.Google Scholar
  19. 19.
    Keady, G., & J. Norbury, On the existence theory for irrotational water waves. Math. Proc. Camb. Phil. Soc., 83 (1978), 137–157.Google Scholar
  20. 20.
    Keady, G., & W. G. Pritchard, Bounds for surface solitary waves. Proc. Camb. Phil. Soc., 76 (1974), 345–358.Google Scholar
  21. 21.
    Korteweg, D. J., & G. DeVries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Phil. Mag., (5) 39 (1895), 422–443.Google Scholar
  22. 22.
    Krasnosel'skii, M. A., Positive solutions of operator equations. Noordhoff, Groningen, 1964.Google Scholar
  23. 23.
    Krasovskii, Yu. P., On the theory of steady state waves of large amplitude. U.S.S.R. Computational Maths. and Math. Phys., 1 (1961), 996–1018.Google Scholar
  24. 24.
    Krasovskii, Yu. P., The existence of aperiodic flows with free boundaries in fluid mechanics. Doklady Akad. Nauk., Vol. 133, 4 (1960), 768–770.Google Scholar
  25. 25.
    Lamb, H., Hydrodynamics. Cambridge University Press, 1932.Google Scholar
  26. 26.
    Lavrentiev, M. A., On the theory of long waves (1943); a contribution to the theory of long waves (1947). Amer. Math. Soc. Translation No. 102 (1954).Google Scholar
  27. 27.
    Leibovich, S., Weakly nonlinear waves in rotating fluids. J. Fluid Mech., 42 (4) (1970), 803–822.Google Scholar
  28. 28.
    Lewy, H., A note on harmonic functions and a hydrodynamical application, Proc. Amer. Math. Soc., 3 (1952), 111–113.Google Scholar
  29. 29.
    Longuet-Higgins, M. S., On the mass, momentum, energy, and circulation of a solitary wave. Proc. R. Soc. Lond. A., 337 (1974), 1–13.Google Scholar
  30. 30.
    Longuet-Higgins, M. S., & J. D. Fenton, On the mass, momentum, energy and circulation of a solitary wave, II. Proc. R. Soc. Lond. A., 340 (1974), 471–493.Google Scholar
  31. 31.
    Longuet-Higgins, M. S., & M. J. H. Fox, Theory of the almost highest wave: the inner solution. J. Fluid Mech., 80 (1977), 721–742.Google Scholar
  32. 32.
    Longuet-Higgins, M. S., & M. J. H. Fox, Theory of the almost highest wave. Part 2. Matching and analytic extension. J. Fluid Mech., 85 (4) (1978), 769–786.Google Scholar
  33. 33.
    McLeod, J. B., Stokes and Krasovskii's conjecture for the wave of greatest height, MRC Technical Summary Report # 2041, University of Wisconsin, Madison (1980).Google Scholar
  34. 34.
    Milne-Thomson, L. M., Theoretical Hydrodynamics. MacMillan, 1977.Google Scholar
  35. 35.
    Miura, R. M., The Korteweg-deVries equation-a survey of results. SIAM Review, 18 (3) (1976), 412–459.PubMedGoogle Scholar
  36. 36.
    Nekrasov, A. I., The exact theory of steady state waves on the surface of a heavy liquid. MRC Technical Summary Report # 813, University of Wisconsin, Madison (1967).Google Scholar
  37. 37.
    Peters, A. S., & J. J. Stoker, Solitary waves in fluids having non-constant density. Comm. Pure Appl. Math., 13 (1960), 115–164.Google Scholar
  38. 38.
    Pritchard, W. G., Solitary waves in rotating fluids. J. Fluid Mech., 42 (1) (1970), 61–83.Google Scholar
  39. 39.
    Rabinowitz, P. H., Some global results for nonlinear eigenvalue problems. J. Functional Anal., 7 (1971), 487–513.Google Scholar
  40. 40.
    Rayleigh, Lord, On Waves. Phil. Mag. (5),i. (1876), 257–279 (Papers, i, 251).Google Scholar
  41. 41.
    Russell, J. S., Report on waves. Rep. 14th meeting of the British Association for the Advancement of Science, John Murray, London (1844), 311–390.Google Scholar
  42. 42.
    Spielvogel, E. R., A variational principle for waves of infinite depth. Arch. Rational Mech. Anal., 39 (1970), 189–205.Google Scholar
  43. 43.
    Starr, V. T., Momentum and energy integrals for gravity waves of finite height. J. Mar. Res., 6 (1947), 175–193.Google Scholar
  44. 44.
    Stokes, G. G., On the theory of oscillatory waves. Math. and Phys. Pap. 1, Cambridge University Press.Google Scholar
  45. 45.
    Ter-Krikorov, A. M., The existence of periodic waves which degenerate into a solitary wave. J. Appl. Maths. Mech., 24 (1960), 930–949.Google Scholar
  46. 46.
    Ter-Krikorov, A. M., Théorie exacte des ondes longues stationnaires dans un liquide hétérogene. J. Mécanique, 2 (1963), 351–376.Google Scholar
  47. 47.
    Toland, J. F., On the existence of a wave of greatest height and Stokes' conjecture. Proc. R. Soc. A., 363 (1978), 469–485.Google Scholar
  48. 48.
    Turner, R. E. L., Transversality and cone maps. Arch. Rational Mech. Anal., 58 (1975), 151–179.Google Scholar
  49. 49.
    Whyburn, G. T., Topological analysis. Princeton University Press, 1958.Google Scholar
  50. 50.
    Yosida, K., Functional analysis. Springer-Verlag, Berlin, 1974.Google Scholar

Copyright information

© Springer-Verlag GmbH & Co 1981

Authors and Affiliations

  • C. J. Amick
    • 1
    • 2
  • J. F. Toland
    • 1
    • 2
  1. 1.Department of MathematicsUniversity of ChicagoUSA
  2. 2.Department of MathematicsUniversity CollegeLondon

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