Open Access
Volume 56, Number 2, March-April 2022
Page(s) 593 - 615
Published online 03 March 2022
  1. S. Alinhac and P. Gérard, Opérateurs pseudo-différentiels et théorème de Nash-Moser. Savoirs Actuels. [Current Scholarship]. InterEditions, Paris; Éditions du Centre National de la Recherche Scientifique (CNRS), Meudon (1991). [Google Scholar]
  2. T.B. Benjamin, J.L. Bona and J.J. Mahony, Model equations for long waves in nonlinear dispersive systems. Philos. Trans. Roy. Soc. London Ser. A 272 (1972) 47–78. [Google Scholar]
  3. J.L. Bona, T. Colin and D. Lannes, Long wave approximations for water waves. Arch. Ration. Mech. Anal. 178 (2005) 373–410. [CrossRef] [MathSciNet] [Google Scholar]
  4. J. Boussinesq, 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 (1872) 55–108. [MathSciNet] [Google Scholar]
  5. C. Burtea, Long time existence results for bore-type initial data for BBM-Boussinesq systems. J. Differ. Equ. 261 (2016) 4825–4860. [CrossRef] [Google Scholar]
  6. C. Burtea, New long time existence results for a class of Boussinesq-type systems. J. Math. Pures Appl. 106 (2016) 203–236. [CrossRef] [MathSciNet] [Google Scholar]
  7. F. Chazel, Influence of bottom topography on long water waves. ESAIM: M2AN 41 (2007) 771–799. [CrossRef] [EDP Sciences] [Google Scholar]
  8. M. Chen, Exact solutions of various Boussinesq systems. Appl. Math. Lett. 11 (1998) 45–49. [CrossRef] [MathSciNet] [Google Scholar]
  9. D. Clamond and D. Dutykh, Fast accurate computation of the fully nonlinear solitary surface gravity waves. Comput. Fluids 84 (2013) 35–38. [CrossRef] [MathSciNet] [Google Scholar]
  10. W. Craig and C. Sulem, Numerical simulation of gravity waves. J. Comput. Phys. 108 (1993) 73–83. [CrossRef] [MathSciNet] [Google Scholar]
  11. W. Craig, C. Sulem and P.-L. Sulem, Nonlinear modulation of gravity waves: a rigorous approach. Nonlinearity 5 (1992) 497–522. [CrossRef] [MathSciNet] [Google Scholar]
  12. O. Darrigol, The spirited horse, the engineer, and the mathematician: water waves in nineteenth-century hydrodynamics. Arch. Hist. Exact Sci. 58 (2003) 21–95. [CrossRef] [MathSciNet] [Google Scholar]
  13. V. Duchêne and S. Israwi, Well-posedness of the Green-Naghdi and Boussinesq–Peregrine systems. Annal. Math. Blaise Pascal 25 (2018) 21–74. [CrossRef] [Google Scholar]
  14. D. Dutykh and D. Clamond, Efficient computation of steady solitary gravity waves. Wave Motion 51 (2014) 86–99. [CrossRef] [MathSciNet] [Google Scholar]
  15. A.E. Green and P.M. Naghdi, A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78 (1976) 237–246. [CrossRef] [Google Scholar]
  16. A.E. Green, N. Laws and P.M. Naghdi, On the theory of water waves. Proc. Roy. Soc. London Ser. A 338 (1974) 43–55. [MathSciNet] [Google Scholar]
  17. M. Haidar, T. El Arwadi and S. Israwi, Explicit solutions and numerical simulations for an asymptotic water waves model with surface tension. J. Appl. Math. Comput. 63 (2020) 655–681. [CrossRef] [MathSciNet] [Google Scholar]
  18. S. Israwi, Derivation and analysis of a new 2d Green-Naghdi system. Nonlinearity 23 (2010) 2889–2904. [CrossRef] [MathSciNet] [Google Scholar]
  19. S. Israwi, Large time existence for 1D Green-Naghdi equations. Nonlinear Anal. 74 (2011) 81–93. [CrossRef] [MathSciNet] [Google Scholar]
  20. S. Israwi and A. Mourad, An explicit solution with correctors for the Green-Naghdi equations. Mediterr. J. Math. 11 (2014) 519–532. [CrossRef] [MathSciNet] [Google Scholar]
  21. T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 41 (1988) 891–907. [CrossRef] [MathSciNet] [Google Scholar]
  22. B. Khorbatly, A remark on the well-posedness of the classical Green-Naghdi system. Math. Methods Appl. Sci. 44 (2021) 14545–14555. [CrossRef] [MathSciNet] [Google Scholar]
  23. B. Khorbatly and S. Israwi, Full justification for the extended Green-Naghdi system for an uneven bottom with/without surface tension. Publ. Res. Inst. Math. Sci. (2022). [Google Scholar]
  24. B. Khorbatly, I. Zaiter and S. Israwi, Derivation and well-posedness of the extended Green-Naghdi equations for flat bottoms with surface tension. J. Math. Phys. 59 (2018) 071501. [CrossRef] [MathSciNet] [Google Scholar]
  25. B. Khorbatly, S. Israwi and T.A.L. Arwadi, An equivalent system to the 2d Green-Naghdi equations. BAU J. - Sci. Tech. hal-02525140, version 1 (2020). [Google Scholar]
  26. D.J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. 39 (1895) 422–443. [CrossRef] [MathSciNet] [Google Scholar]
  27. D. Lannes, Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators. J. Funct. Anal. 232 (2006) 495–539. [CrossRef] [MathSciNet] [Google Scholar]
  28. D. Lannes, The water waves problem, In Vol. 188 of Mathematical Surveys and Monographs. American Mathematical Society. Mathematical analysis and asymptotics, Providence, RI (2013). [CrossRef] [Google Scholar]
  29. D. Lannes and P. Bonneton, Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation. Phys. Fluids 21 (2009) 016601. [CrossRef] [Google Scholar]
  30. F. Linares, D. Pilod and J.-C. Saut, Well-posedness of strongly dispersive two-dimensional surface wave Boussinesq systems. SIAM J. Math. Anal. 44 (2012) 4195–4221. [CrossRef] [MathSciNet] [Google Scholar]
  31. R. Lteif and S. Gerbi, A new class of higher-ordered/extended boussinesq system for efficient numerical simulations by splitting operators. Preprint (2021). [Google Scholar]
  32. Y. Matsuno, Hamiltonian formulation of the extended Green-Naghdi equations. Phys. D 301/302 (2015) 1–7. [CrossRef] [Google Scholar]
  33. Y. Matsuno, Hamiltonian structure for two-dimensional extended Green-Naghdi equations. Proc. Roy. Soc. A: Math. Phys. Eng. Sci. 472 (2016) 20160127. [Google Scholar]
  34. M. Ming, J.C. Saut and P. Zhang, Long-time existence of solutions to Boussinesq systems. SIAM J. Math. Anal. 44 (2012) 4078–4100. [CrossRef] [MathSciNet] [Google Scholar]
  35. J.W.S. Rayleigh, On waves. London, Edinburgh, Dublin Philos. Mag. J. Sci. 1 (1876) 257–279. [CrossRef] [Google Scholar]
  36. J.-C. Saut and L. Xu, The Cauchy problem on large time for surface waves Boussinesq systems. J. Math. Pures Appl. 97 (2012) 635–662. [CrossRef] [MathSciNet] [Google Scholar]
  37. J.-C. Saut and L. Xu, Long time existence for a strongly dispersive boussinesq system. SIAM J. Numer. Anal. 52 (2020). [Google Scholar]
  38. J.-C. Saut and L. Xu, Long time existence for the boussinesq-full dispersion systems. J. Differ. Equ. 269 (2020) 2627–2663. [CrossRef] [Google Scholar]
  39. J.-C. Saut and L. Xu, Long time existence for a two-dimensional strongly dispersive boussinesq system. Commun. Partial Differ. Equ. 46 (2021) 2057–2087. [CrossRef] [Google Scholar]
  40. J.-C. Saut, C. Wang and L. Xu, The Cauchy problem on large time for surface-waves-type Boussinesq systems II. SIAM J. Math. Anal. 49 (2017) 2321–2386. [CrossRef] [MathSciNet] [Google Scholar]
  41. F. Serre, Contribution à l’étude des écoulements permanents et variables dans les canaux. La Houille Blanche 39 (1953) 374–388. [CrossRef] [EDP Sciences] [Google Scholar]
  42. C.H. Su and C.S. Gardner, Korteweg-de Vries equation and generalizations. III. Derivation of the Korteweg-de Vries equation and Burgers equation. J. Math. Phys. 10 (1969) 536–539. [CrossRef] [Google Scholar]
  43. M. Tanaka, The stability of solitary waves. Phys. Fluids 29 (1986) 650–655. [CrossRef] [MathSciNet] [Google Scholar]
  44. V.E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9 (1968) 190–194. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you