Journal of Mathematical Fluid Mechanics

, Volume 20, Issue 3, pp 1059–1091 | Cite as

Solitary Wave Solutions to a Class of Modified Green–Naghdi Systems

  • Vincent Duchêne
  • Dag Nilsson
  • Erik WahlénEmail author
Open Access


We provide the existence and asymptotic description of solitary wave solutions to a class of modified Green–Naghdi systems, modeling the propagation of long surface or internal waves. This class was recently proposed by Duchêne et al. (Stud Appl Math 137:356–415, 2016) in order to improve the frequency dispersion of the original Green–Naghdi system while maintaining the same precision. The solitary waves are constructed from the solutions of a constrained minimization problem. The main difficulties stem from the fact that the functional at stake involves low order non-local operators, intertwining multiplications and convolutions through Fourier multipliers.



V. Duchêne was partially supported by the Agence Nationale de la Recherche (project ANR-13-BS01-0003-01 DYFICOLTI). D. Nilsson and E. Wahlén were supported by the Swedish Research Council (Grant No. 621-2012-3753).


  1. 1.
    Aceves-Sánchez, P., Minzoni, A.A., Panayotaros, P.: Numerical study of a nonlocal model for water-waves with variable depth. Wave Motion 50, 80–93 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Angulo Pava, J.: Nonlinear Dispersive Equations. Existence and Stability of Solitary and Periodic Travelling Wave Solutions, Vol. 156 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2009)zbMATHGoogle Scholar
  3. 3.
    Arnesen, M.N.: Existence of solitary-wave solutions to nonlocal equations. Discrete Contin. Dyn. Syst. 36, 3483–3510 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 343. Springer, Heidelberg (2011)zbMATHGoogle Scholar
  5. 5.
    Benzoni-Gavage, S., Serre, D.: Multidimensional Hyperbolic Partial Differential Equations. First-Order Systems and Applications. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2007)zbMATHGoogle Scholar
  6. 6.
    Bona, J.L., Souganidis, P.E., Strauss, W.A.: Stability and instability of solitary waves of Korteweg-de Vries type. Proc. R. Soc. Lond. Ser. A 411, 395–412 (1987)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Boussinesq, J.: Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. 17, 55–108 (1872)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Buffoni, B.: Existence and conditional energetic stability of capillary-gravity solitary water waves by minimisation. Arch. Ration. Mech. Anal. 173, 25–68 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Camassa, R., Choi, W., Michallet, H., Rusas, P.-O., Sveen, J.K.: On the realm of validity of strongly nonlinear asymptotic approximations for internal waves. J. Fluid Mech. 549, 1–23 (2006)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Carter, J.D.: Bidirectional Whitham Equations as Models of Waves on Shallow Water. arXiv:1705.06503v1
  12. 12.
    Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. The International Series of Monographs on Physics. Clarendon Press, Oxford (1961)Google Scholar
  13. 13.
    Chemin, J.-Y.: Perfect Incompressible Fluids, Vol. 14 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York (1998). Translated from the 1995 French original by Isabelle Gallagher and Dragos IftimieGoogle Scholar
  14. 14.
    Chen, M.: Exact solutions of various Boussinesq systems. Appl. Math. Lett. 11, 45–49 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chen, M., Nguyen, N.V., Sun, S.-M.: Solitary-wave solutions to Boussinesq systems with large surface tension. Discrete Contin. Dyn. Syst. 26, 1153–1184 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Chen, M., Nguyen, N.V., Sun, S.-M.: Existence of traveling-wave solutions to Boussinesq systems. Differ. Integral Equ. 24, 895–908 (2011)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Choi, W., Camassa, R.: Fully nonlinear internal waves in a two-fluid system. J. Fluid Mech. 396, 1–36 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Clamond, D., Dutykh, D.: Fast accurate computation of the fully nonlinear solitary surface gravity waves. Comput. Fluids 84, 35–38 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust-Region Methods. MPS/SIAM Series on Optimization, Society for Industrial and Applied Mathematics (SIAM). Mathematical Programming Society (MPS), Philadelphia (2000)Google Scholar
  20. 20.
    Darrigol, O.: The spirited horse, the engineer, and the mathematician: water waves in nineteenth-century hydrodynamics. Arch. Hist. Exact Sci. 58, 21–95 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Djordjevic, V.D., Redekopp, L.G.: The fission and disintegration of internal solitary waves moving over two-dimensional topography. J. Phys. Oceanogr. 8, 1016–1024 (1978)ADSCrossRefGoogle Scholar
  22. 22.
    Duchêne, V.: On the rigid-lid approximation for two shallow layers of immiscible fluids with small density contrast. J. Nonlinear Sci. 24, 579–632 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Duchêne, V., Israwi, S., Talhouk, R.: A new class of two-layer Green-Naghdi systems with improved frequency dispersion. Stud. Appl. Math. 137, 356–415 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ehrnström, M., Groves, M.D., Wahlén, E.: On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type. Nonlinearity 25, 2903–2936 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Groves, M.D., Wahlén, E.: Existence and conditional energetic stability of solitary gravity-capillary water waves with constant vorticity. Proc. R. Soc. Edinb. Sect. A 145, 791–883 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Grue, J., Jensen, A., Rusås, P.-O., Sveen, J.K.: Properties of large-amplitude internal waves. J. Fluid Mech. 380, 257–278 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Helfrich, K .R., Melville, W .K.: Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38, 395–425 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Hur, V., Pandey, A.: Modulational instability in a full-dispersion shallow water model. arXiv:1608.04685v1
  29. 29.
    Jackson, C.R.: An atlas of internal solitary-like waves and their properties (2004).
  30. 30.
    Klein, C., Linares, F., Pilod, D., Saut, J.-C.: On Whitham and related equations. Studies in Applied Mathematics (2017).
  31. 31.
    Korteweg, D.J., De Vries, G.: On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. 5, 422–443 (1895)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Lannes, D.: The Water Waves Problem, Vol. 188 of Mathematical Surveys and Monographs. Mathematical Analysis and Asymptotics. American Mathematical Society, Providence (2013)Google Scholar
  33. 33.
    Li, Y.A.: Hamiltonian structure and linear stability of solitary waves of the Green–Naghdi equations. J. Nonlinear Math. Phys. 9, 99–105 (2002). Recent advances in integrable systems (Kowloon, 2000)Google Scholar
  34. 34.
    Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact case. I. Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 109–145 (1984)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Mal’tseva, Z.L.: Unsteady long waves in a two-layer fluid. Dinamika Sploshn. Sredy 93, 96–110 (1989)MathSciNetGoogle Scholar
  36. 36.
    Métivier, G.: Para-Differential Calculus and Applications to the Cauchy Problem for Nonlinear Systems. Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series, vol. 5. Edizioni della Normale, Pisa (2008)Google Scholar
  37. 37.
    Michallet, H., Barthélemy, E.: Experimental study of interfacial solitary waves. J. Fluid Mech. 366, 159–177 (1998)ADSCrossRefzbMATHGoogle Scholar
  38. 38.
    Miyata, M.: Long internal waves of large amplitude. In: Nonlinear Water Waves: IUTAM Symposium, pp. 399–405. Springer, Tokyo (1987)Google Scholar
  39. 39.
    Moldabayev, D., Kalisch, H., Dutykh, D.: The Whitham equation as a model for surface water waves. Phys. D 309, 99–107 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Ostrovsky, L.A., Stepanyants, Y.A.: Do internal solitons exist in the ocean? Rev. Geophys. 27, 293–310 (1989)ADSCrossRefGoogle Scholar
  41. 41.
    Rayleigh, J.W.S.: On waves. Philos. Mag. 1, 251–271 (1876)zbMATHGoogle Scholar
  42. 42.
    Serre, F.: Contribution à l’étude des écoulements permanents et variables dans les canaux, pp. 830–872. La Houille Blanche, Grenoble (1953)Google Scholar
  43. 43.
    Struwe, M.: Variational Methods, Vol. 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 4th edn. Springer, Berlin (2008)Google Scholar
  44. 44.
    Trefethen, L.N.: Spectral methods in MATLAB, Vol. 10 of Software, Environments, and Tools. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2000)CrossRefGoogle Scholar

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.IRMAR - UMR 6625University of Rennes 1, CNRSRennesFrance
  2. 2.Centre for Mathematical SciencesLund UniversityLundSweden

Personalised recommendations