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References

  1. B. Alvarez-Samaniego and D. Lannes, Large Time Existence and3D Asymptotics for Water-Waves, Preprint, arXiv:math/0702015v1, http://arxiv.org/abs/math/0702015v1.

    Google Scholar 

  2. B. Alvarez-Samaniego and D. Lannes, A Nash–Moser theorem with parameters for evolution equations, Indiana Univ. Math. J. [To appear]

    Google Scholar 

  3. W. Ben Youssef and D. Lannes, The long wave limit for a general class of 2D quasilinear hyperbolic problems, Commun. Partial Differ. Equations 27(2002), 979-1020.

    Article  MATH  Google Scholar 

  4. J. L. Bona, T. Colin, and D. Lannes, Long waves approximations for water waves, Arch. Ration. Mech. Anal. 178(2005), NO.3, 373-410.

    Article  MATH  MathSciNet  Google Scholar 

  5. J. L. Bona, M. Chen, and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory, J. Nonlinear Sci. 12(2002), no. 4, 283- 318.

    Google Scholar 

  6. F. Chazel, In.uence of topography on water waves, M2AN. [To appear]

    Google Scholar 

  7. D. Coutand and S. Shkoller, Well-posedness of the free-surface incompressible Euler equations with or without surface tension, http://arxiv.org/ps/math.AP/0511236.

    Google Scholar 

  8. W. Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits, Commun. Partial Differ. Equations 10(1985), no. 8, 787-1003.

    Article  MATH  MathSciNet  Google Scholar 

  9. W. Craig, Nonstrictly hyperbolic nonlinear systems, Math. Ann. 277(1987), no. 2, 213-232.

    Article  MATH  MathSciNet  Google Scholar 

  10. W.Craig, U. Schanz, and C. Sulem, The modulational regime of threedimensional water waves and the Davey–Stewartson system, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 14(1997), no. 5, 615-667.

    Article  MATH  MathSciNet  Google Scholar 

  11. W. Craig, C. Sulem, and P.-L. Sulem, Nonlinear modulation of gravity waves: a rigorous approach, Nonlinearity 5(1992), no. 2, 497-522.

    Article  MATH  MathSciNet  Google Scholar 

  12. M. W. Dingemans, Water Wave Propagation over Uneven Bottoms. Part 2: Nonlinear Wave Propagation, Advanced Series on Ocean Engineering, Vol. 13, World Sci. Publ., 1997.

    Google Scholar 

  13. T. Iguchi, A long wave approximation for capillary-gravity waves and an e.ect of the bottom, Commun. Partial Differ. Equations 32(2007), 37-85.

    Article  MATH  MathSciNet  Google Scholar 

  14. T. Kano, L’équation de Kadomtsev–Petviashvili approchant les ondes longues de surface de l’eau en écoulement trois-dimensionnel, In: Patterns and Waves, North-Holland, Amsterdam, 1986, pp. 431-444.

    Google Scholar 

  15. T. Kano and T. Nishida, Sur les ondes de surface de l’eau avec une justification mathematique des équations des ondes en eau peu profonde, J. Math. Kyoto Univ. 19(1979), no. 2, 335-370.

    MATH  MathSciNet  Google Scholar 

  16. T. Kano and T. Nishida, A mathematical justification for Korteweg-de Vries equation and Boussinesq equation of water surface waves, Osaka J. Math. 23(1986), no. 2, 389-413.

    MATH  MathSciNet  Google Scholar 

  17. D. Lannes, Well-posedness of the water-waves equations, J. Am. Math. Soc. 18(2005), 605-654.

    Article  MATH  MathSciNet  Google Scholar 

  18. D. Lannes and J.-C. Saut, Weakly transverse Boussinesq systems and the KP approximation, Nonlinearity 19(2006), 2853-2875.

    Article  MATH  MathSciNet  Google Scholar 

  19. H. Lindblad,Well-posedness for the linearized motion of an incompressible liquid with free surface boundary, Commun. Pure Appl. Math. 56(2003), no. 2, 153-197.

    Article  MATH  MathSciNet  Google Scholar 

  20. H. Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary, Ann. Math. (2) 162(2005), no. 1, 109-194.

    Article  MATH  MathSciNet  Google Scholar 

  21. Y. Matsuno, Nonlinear evolution of surface gravity waves on fluid of finite depth, Phys. Rev. Lett. 69(1992), no. 4, 609-611.

    Article  MATH  MathSciNet  Google Scholar 

  22. V. I. Nalimov, The Cauchy-Poisson problem[in Russian], Dinamika Sploshn. Sred. 18(1974), 104-210, 254.

    Google Scholar 

  23. L.V. Ovsjannikov, Cauchy problem in a scale of Banach spaces and its application to the shallow water theory justification, In: Appl. Methods Funct. Anal. Probl. Mech. Lect. Notes Math. 503, 1976, pp. 426-437.

    Article  MathSciNet  Google Scholar 

  24. G. Schneider and C. E. Wayne, The long-wave limit for the water wave problem. I. The case of zero surface tension, Commun. Pure Appl. Math. 53(2000), no. 12, 1475-1535.

    Article  Google Scholar 

  25. G. Schneider and C. E. Wayne, The rigorous approximation of longwavelength capillary-gravity waves, Arch. Ration. Mech. Anal. 162(2002), no. 3, 247-285.

    Article  MATH  MathSciNet  Google Scholar 

  26. J. Shatah and C. Zeng, Geometry and a priori estimates for free boundary problems of the Euler’s equation, http://arxiv.org/abs/math.AP/0608428

    Google Scholar 

  27. J. D. Wright, Corrections to the KdV approximation for water waves, SIAM J. Math. Anal. 37(2005), no. 4, 1161-1206.

    Article  MathSciNet  Google Scholar 

  28. S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in2-D, Invent. Math. 130(1997), no. 1, 39-72.

    Google Scholar 

  29. S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Am. Math. Soc. 12(1999), no. 2, 445-495.

    Article  MATH  Google Scholar 

  30. H. Yosihara, Gravity waves on the free surface of an incompressible perfect fluid of finite depth, Publ. Res. Inst. Math. Sci. 18(1982), no. 1, 49-96.

    MATH  MathSciNet  Google Scholar 

  31. H. Yosihara, Capillary-gravity waves for an incompressible ideal fluid. J. Math. Kyoto Univ. 23(1983), no. 4, 649-694.

    MATH  MathSciNet  Google Scholar 

  32. V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys. 2(1968), 190-194.

    Google Scholar 

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Lannes, D. (2008). Justifying Asymptotics for 3D Water–Waves. In: Bardos, C., Fursikov, A. (eds) Instability in Models Connected with Fluid Flows II. International Mathematical Series, vol 7. Springer, New York, NY. https://doi.org/10.1007/978-0-387-75219-8_1

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