Archive for Rational Mechanics and Analysis

, Volume 228, Issue 3, pp 773–820 | Cite as

A Variational Reduction and the Existence of a Fully Localised Solitary Wave for the Three-Dimensional Water-Wave Problem with Weak Surface Tension



Fully localised solitary waves are travelling-wave solutions of the three- dimensional gravity–capillary water wave problem which decay to zero in every horizontal spatial direction. Their existence has been predicted on the basis of numerical simulations and model equations (in which context they are usually referred to as ‘lumps’), and a mathematically rigorous existence theory for strong surface tension (Bond number \({\beta}\) greater than \({\frac{1}{3}}\)) has recently been given. In this article we present an existence theory for the physically more realistic case \({0 < \beta < \frac{1}{3}}\). A classical variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle featuring a perturbation of the functional associated with the Davey–Stewartson equation. A nontrivial critical point of the reduced functional is found by minimising it over its natural constraint set.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ablowitz M.J., Segur H.: On the evolution of packets of water waves. J. Fluid Mech. 92, 691–715 (1979)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Benci V., Cerami G.: Positive solutions of some nonlinear elliptic problems in exterior domains. Arch. Rat. Mech. Anal. 99, 283–300 (1987)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Benjamin T.B., Olver P.J.: Hamiltonian structure, symmetries and conservation laws for water waves. J. Fluid Mech. 125, 137–185 (1982)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Buffoni, B: Existence of fully localised water waves with weak surface tension. Mathematisches Forschungsinstitut Oberwolfach, Report no 19/2015, 1037–1039, 2015.Google Scholar
  5. 5.
    Buffoni B., Groves M.D., Sun S.M., Wahlén E.: Existence and conditional energetic stability of three-dimensional fully localised solitary gravity–capillary water waves. J. Differ. Equ. 254, 1006–1096 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cipolatti R.: On the existence of standing waves for a Davey–Stewartson system. Commun. Part. Differ. Equ. 17, 967–988 (1992)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Craig W.: Water waves, Hamiltonian systems and Cauchy integrals. In Microlocal Analysis and Nonlinear Waves (Eds. Beals M., Melrose R. B. and Rauch J.) Springer, New York, 37–45, 1991.Google Scholar
  8. 8.
    Dias F., Kharif C.: Nonlinear gravity and capillary–gravity waves. Ann. Rev. Fluid Mech. 31, 301–346 (1999)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Djordjevic V.D., Redekopp L.G.: On two-dimensional packets of capillary–gravity waves. J. Fluid Mech. 79, 703–714 (1977)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ekeland I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Groves M.D., Sun S.-M.: Fully localised solitary-wave solutions of the three-dimensional gravity–capillary water-wave problem. Arch. Rat. Mech. Anal. 188, 1–91 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Hörmander L.: Lectures on Nonlinear Hyperbolic Differential Equations. Springer, Heidelberg (1997)MATHGoogle Scholar
  13. 13.
    Kadomtsev B.B., Petviashvili V.I.: On the stability of solitary waves in weakly dispersing media. Sov. Phys. Dokl. 15, 539–541 (1970)ADSMATHGoogle Scholar
  14. 14.
    Lions P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case, part 1. Ann. Inst. Henri Poincaré Anal. Non Linéaire 1, 109–145 (1984)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Lions P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case, part 2. Ann. Inst. Henri Poincaré Anal. Non Linéaire 1, 223–283 (1984)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Papanicolaou G.C., Sulem C., Sulem P.L., Wang X.P.: The focusing singularity of the Davey–Stewartson equations for gravity–capillary surface waves. Physica D 72, 61–86 (1994)ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Parau, E.I., Vanden-Broeck, J.-M., Cooker, M.J.: Three-dimensional gravity–capillary solitary waves in water of finite depth and related problems. Phys. Fluids 17, 122101, 2005.Google Scholar
  18. 18.
    Willem M.: Minimax Theorems. Birkhäuser, Boston (1996)CrossRefMATHGoogle Scholar
  19. 19.
    Zakharov V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190–194 (1968)ADSCrossRefGoogle Scholar
  20. 20.
    Zakharov V.E., Kuznetsov E.A.: Three-dimensional solitons. Zh. Eksp. Teor. Fiz. 66, 594–597 (1974)ADSGoogle Scholar
  21. 21.
    Zakharov V.E., Kuznetsov E.A.: Hamiltonian formalism for systems of hydrodynamic type. Sov. Sci. Rev. Sec. C: Math. Phys. Rev. 4, 167–220 (1984)MathSciNetMATHGoogle Scholar
  22. 22.
    Zakharov V.E., Kuznetsov E.A.: Hamiltonian formalism for nonlinear waves. Physics-Uspekhi 40, 1087–1116 (1997)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Section de mathématiques, Station 8Ecole Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2.Fachrichtung MathematikUniversität des SaarlandesSaarbrückenGermany
  3. 3.Department of Mathematical SciencesLoughborough UniversityLoughboroughUK
  4. 4.Centre for Mathematical SciencesLund UniversityLundSweden

Personalised recommendations