Existence Proof for Weakly Nonlinear Water Waves

Part of the Theoretical and Mathematical Physics book series (TMP)


The availability of closed analytic solutions for nonlinear water waves is the rare exception; some examples are the Crapper capillary wave from the last section, as well as cnoidal waves and the Gerstner wave for free surface waves.


  1. Arnold, V.I. 1963. Proof of a theorem by A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian. Uspekhi Matematicheskikh Nauk 18: 13. Russian Mathematical Surveys 18: 9.Google Scholar
  2. Buffoni, B., and J. Toland. 2003. Analytic theory of global bifurcation. Princeton: Princeton University Press.Google Scholar
  3. Dieudonné, J. 1969. Foundations of modern analysis. New York: Academic Press.Google Scholar
  4. Dunford, N., and J.T. Schwartz. 1958. Linear operators. Part I: General theory. New York: Interscience Publishers.Google Scholar
  5. Fredholm, I. 1903. Sur une classe d’équations fonctionnelles. Acta Mathematica 27: 365.CrossRefMathSciNetGoogle Scholar
  6. Hellinger, E., and O. Toeplitz. 1927. Integralgleichungen und Gleichungen mit unendlich vielen Unbekannten. In Encyklopädie der mathematischen Wissenschaften, vol. 2-3-2, 1335. Leipzig: Teubner (digital at GDZ Göttingen).Google Scholar
  7. Hutson, V., and J.S. Pym. 1980. Applications of functional analysis and operator theory. London: Academic Press.Google Scholar
  8. Kolmogorov, A.N. 1954. The general theory of dynamical systems and classical mechanics. In Proceedings of the international congress of mathematicians, Amsterdam, vol. 1, 315. North Holland, Amsterdam, 1957 (in Russian). English translation as Appendix in R.H. Abraham and J.E. Marsden, Foundations of mechanics, 2nd ed. Benjamin/Cummings, 1978.Google Scholar
  9. Krasnoselskii, M.A. 1964a. Positive solutions of operator equations. Groningen: P. Noordhoff Ltd.Google Scholar
  10. Krasnoselskii, M.A. 1964b. Topological methods in the theory of nonlinear integral equations. New York: The Macmillan Company; Oxford: Pergamon Press.Google Scholar
  11. Krasnoselskii, M.A., et al. 1972. Approximate solutions of operator equations. Groningen: Wolters-Noordhoff Publishing.Google Scholar
  12. Krasovskii, Yu.P. 1961. On the theory of steady-state waves of finite amplitude. U.S.S.R. Computational Mathematics and Mathematical Physics 1: 996.Google Scholar
  13. Levandosky, J. 2003. Partial differential equations of applied mathematics. Course Notes for Math 220B.
  14. Levi-Civita, T. 1925. Détermination rigoureuse des ondes permanentes d’ampleur finie. Mathematische Annalen 93: 264.CrossRefMathSciNetGoogle Scholar
  15. Lichtenstein, L. 1931. Vorlesungen über einige Klassen nichtlinearer Integralgleichungen und Integro-Differentialgleichungen. Berlin: Springer.Google Scholar
  16. Littman, W. 1957. On the existence of periodic waves near critical speed. Communications on Pure and Applied Mathematics 10: 241.CrossRefMathSciNetGoogle Scholar
  17. Moser, J. 1962. On invariant curves of area-preserving mappings of an annulus. Nachrichten der Akademie der Wissenschaften in Göttingen, Mathematisch-Physikalische Klasse II 1: 1.zbMATHMathSciNetGoogle Scholar
  18. Nekrasov, A.I. 1921. On steady waves. Izv. Ivanovo-Voznesenskogo politekhn. in-ta 3. Translated by D.V. Thampuran: The exact theory of steady state waves on the surface of a heavy liquid. 1967, Technical Summary Report No. 813, Mathematics Research Center, United States Army, University of Wisconsin, ed. C.W. Cryer. Available at Hathi Trust Digital Library.
  19. Neumann, C. 1877. Untersuchungen über das logarithmische und Newtonsche Potential. Leipzig: Teubner.Google Scholar
  20. Picard, É. 1910. Sur un théorème général relatif aux équations intégrales de première espèce et sur quelques problèmes de physique mathématique. Rendiconti del Circolo Matematico di Palermo 29: 79.CrossRefGoogle Scholar
  21. Schmidt, E. 1907. Zur Theorie der linearen und nichtlinearen Integralgleichungen. Mathematische Annalen 63: 433 (part 1) and 64: 161 (part 2).Google Scholar
  22. Schmidt, E. 1908. Zur Theorie der linearen und nichtlinearen Integralgleichungen. Mathematische Annalen 65: 370 (part 3).Google Scholar
  23. Schmidt, E. 1910. Bemerkung zur Potentialtheorie. Mathematische Annalen 68: 107.CrossRefMathSciNetGoogle Scholar
  24. Schwartz, J.T. 1969. Nonlinear functional analysis. New York: Gordon and Breach.Google Scholar
  25. Smirnow, W.I. 1988. Lehrgang der höheren Mathematik, part IV/1. Berlin: Deutscher Verlag der Wissenschaften.Google Scholar
  26. Sternberg, W. 1925. Potentialtheorie. I. Die Elemente der Potentialtheorie. Berlin: Walter de Gruyter & Co.Google Scholar
  27. Stoker, J.J. 1957. Water waves. The mathematical theory with applications. New York: Interscience Publishers.Google Scholar
  28. Toland, J.F. 1996. Stokes waves. Topological Methods in Nonlinear Analysis 7: 1.CrossRefMathSciNetGoogle Scholar
  29. Wainberg, M.M., and W.A. Trenogin. 1973. Theorie der Lösungsverzweigung bei nichtlinearen Gleichungen. Berlin: Akademie-Verlag.Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut für Physik und AstronomieUniversität PotsdamPotsdamGermany

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