M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, American Journal of Physics, vol.34, issue.2, p.22, 1972.
DOI : 10.1119/1.1972842

G. A. Athanassoulis and K. A. Belibassakis, A consistent coupled-mode theory for the propagation of small-amplitude water waves over variable bathymetry regions, Journal of Fluid Mechanics, vol.389, pp.275-301, 1999.
DOI : 10.1017/S0022112099004978

A. M. Balk, A Lagrangian for water waves, Physics of Fluids, vol.8, issue.2, pp.416-419, 1996.
DOI : 10.1063/1.868795

T. B. Benjamin, J. L. Bona, and J. J. Mahony, Model Equations for Long Waves in Nonlinear Dispersive Systems, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.272, issue.1220, pp.47-78, 1972.
DOI : 10.1098/rsta.1972.0032

J. L. Bona, M. Chen, and J. Saut, Boussinesq Equations and Other Systems for Small-Amplitude Long Waves in Nonlinear Dispersive Media. I: Derivation and Linear Theory, Journal of Nonlinear Science, vol.12, issue.4, pp.283-318, 2002.
DOI : 10.1007/s00332-002-0466-4

J. L. Bona, M. Chen, and J. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: II. The nonlinear theory, Nonlinearity, vol.17, issue.3, pp.925-952, 2004.
DOI : 10.1088/0951-7715/17/3/010

J. V. Boussinesq, Théorie de l'intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire, C.R. Acad. Sci. Paris Sér. A-B, vol.72, pp.755-759, 1871.

T. J. Bridges, Periodic Patterns, Linear Instability, Symplectic Structure and Mean-Flow Dynamics for Three-Dimensional Surface Waves, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.354, issue.1707, pp.533-574, 1996.
DOI : 10.1098/rsta.1996.0019

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Physical Review Letters, vol.71, issue.11, pp.1661-1664, 1993.
DOI : 10.1103/PhysRevLett.71.1661

A. Constantin, D. H. Sattinger, and W. Strauss, Variational formulations for steady water waves with vorticity, Journal of Fluid Mechanics, vol.548, issue.-1, pp.151-163, 2006.
DOI : 10.1017/S0022112005007469

W. Craig and M. D. Groves, Hamiltonian long-wave approximations to the water-wave problem, Wave Motion, vol.19, issue.4, pp.367-389, 1994.
DOI : 10.1016/0165-2125(94)90003-5

W. Craig and C. Sulem, Numerical Simulation of Gravity Waves, Journal of Computational Physics, vol.108, issue.1, pp.73-83, 1993.
DOI : 10.1006/jcph.1993.1164

A. D. Craik, THE ORIGINS OF WATER WAVE THEORY, Annual Review of Fluid Mechanics, vol.36, issue.1, pp.1-28, 2004.
DOI : 10.1146/annurev.fluid.36.050802.122118

A. Degasperis and M. Procesi, Asymptotic integrability, chapter Asymptotic, World Scientific, vol.4, pp.23-37, 1999.

F. Dias and T. J. Bridges, The numerical computation of freely propagating time-dependent irrotational water waves, Fluid Dynamics Research, vol.38, issue.12, pp.803-830, 2006.
DOI : 10.1016/j.fluiddyn.2005.08.007

V. A. Dougalis and D. E. Mitsotakis, Theory and Numerical Analysis of Boussinesq Systems, Effective Computational Methods in Wave Propagation, pp.63-110, 2008.
DOI : 10.1201/9781420010879.ch3

URL : https://hal.archives-ouvertes.fr/hal-00407927

K. B. Dysthe, Note on a modification to the nonlinear Schrödinger equation for application to deep water, Proc. R. Soc. Lond. A, pp.105-114, 1979.

C. Eckart, Variation Principles of Hydrodynamics, Physics of Fluids, vol.3, issue.3, pp.421-427, 1960.
DOI : 10.1063/1.1706053

J. D. Fenton, NUMERICAL METHODS FOR NONLINEAR WAVES, Adv. Coastal Ocean Engng, vol.5, pp.241-324, 1999.
DOI : 10.1142/9789812797544_0005

D. Fructus, D. Clamond, O. Kristiansen, and J. Grue, An efficient model for three-dimensional surface wave simulations, Journal of Computational Physics, vol.205, issue.2, pp.665-685, 2005.
DOI : 10.1016/j.jcp.2004.11.027

F. Geniet, Large amplitude surface waves in infinite depth, a variational approach, p.20, 2003.

A. E. Green, N. Laws, and P. M. Naghdi, On the Theory of Water Waves, Proc. R. Soc. Lond. A, pp.43-55, 1974.
DOI : 10.1098/rspa.1974.0072

A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, Journal of Fluid Mechanics, vol.338, issue.02, pp.237-246, 1976.
DOI : 10.1017/S0022112076002425

J. Grue, D. Clamond, M. Huseby, and A. Jensen, Kinematics of extreme waves in deep water, Applied Ocean Research, vol.25, issue.6, pp.355-366, 2003.
DOI : 10.1016/j.apor.2004.03.001

A. Jensen, D. Clamond, M. Huseby, and J. Grue, On local and convective accelerations in steep wave events, Ocean Engineering, vol.34, issue.3-4, pp.426-435, 2007.
DOI : 10.1016/j.oceaneng.2006.03.013

M. Kameyama, A. Kageyama, and T. Sato, Multigrid iterative algorithm using pseudo-compressibility for three-dimensional mantle convection with strongly variable viscosity, Journal of Computational Physics, vol.206, issue.1, pp.162-181, 2005.
DOI : 10.1016/j.jcp.2004.11.030

D. J. Kaup, A Higher-Order Water-Wave Equation and the Method for Solving It, Progress of Theoretical Physics, vol.54, issue.2, pp.396-408, 1975.
DOI : 10.1143/PTP.54.396

J. W. Kim, K. J. Bai, R. C. Ertekin, and W. C. Webster, A derivation of the Green-Naghdi equations for irrotational flows, Journal of Engineering Mathematics, vol.40, issue.1, pp.17-42, 2001.
DOI : 10.1023/A:1017541206391

J. T. Kirby, Gravity Waves in Water of Finite Depth, pp.55-125, 1997.

R. A. Kraenkel, J. Leon, and M. A. Manna, Theory of small aspect ratio waves in deep water, Physica D: Nonlinear Phenomena, vol.211, issue.3-4, pp.377-390, 2005.
DOI : 10.1016/j.physd.2005.09.001

URL : https://hal.archives-ouvertes.fr/in2p3-00149077

B. A. Kupershmidt, Mathematics of dispersive water waves, Communications in Mathematical Physics, vol.1, issue.1, pp.51-73, 1985.
DOI : 10.1007/BF01466593

J. Lagrange, Mémoire sur la théorie du mouvement des fluides, Nouv. Mém. Acad. Berlin, issue.8, p.1781

D. Lewis, J. Marsden, R. Montgomery, and T. Ratiu, The Hamiltonian structure for dynamic free boundary problems, Physica D: Nonlinear Phenomena, vol.18, issue.1-3, pp.391-404, 1986.
DOI : 10.1016/0167-2789(86)90207-1

Y. A. Li, Hamiltonian Structure and Linear Stability of Solitary Waves of the Green-Naghdi Equations, Journal of Nonlinear Mathematical Physics, vol.284, issue.sup1, pp.99-105, 2002.
DOI : 10.2991/jnmp.2002.9.s1.9

J. C. Luke, A variational principle for a fluid with a free surface, Journal of Fluid Mechanics, vol.125, issue.02, pp.375-397, 1967.
DOI : 10.1007/BF01449125

Q. Ma, Advances in numerical simulation of nonlinear water waves, In Adv. Coastal Ocean Engng. World Scientific, vol.11, p.24, 2010.
DOI : 10.1142/7087

P. A. Madsen, H. B. Bingham, and H. A. Schaffer, Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves: derivation and analysis, Proc. R. Soc. Lond. A, pp.1075-1104, 2003.
DOI : 10.1098/rspa.2002.1067

P. A. Madsen and H. A. Schaffer, A REVIEW OF BOUSSINESQ-TYPE EQUATIONS FOR SURFACE GRAVITY WAVES, Adv. Coastal & Ocean Engin, vol.5, issue.8, pp.1-94, 1999.
DOI : 10.1142/9789812797544_0001

C. C. Mei, The applied dynamics of water waves, World Scientific, issue.4, 1989.

J. W. Miles, On Hamilton's principle for surface waves, Journal of Fluid Mechanics, vol.43, issue.01, pp.153-158, 1977.
DOI : 10.1017/S0022112077001104

P. J. Morrison, Hamiltonian description of the ideal fluid, Reviews of Modern Physics, vol.70, issue.2, pp.467-521, 1998.
DOI : 10.1103/RevModPhys.70.467

O. Nwogu, Alternative Form of Boussinesq Equations for Nearshore Wave Propagation, Journal of Waterway, Port, Coastal, and Ocean Engineering, vol.119, issue.6, pp.618-638, 1993.
DOI : 10.1061/(ASCE)0733-950X(1993)119:6(618)

P. J. Olver, Hamiltonian perturbation theory and water waves, Contemp. Math, vol.28, pp.231-249, 1984.
DOI : 10.1090/conm/028/751987

P. J. Olver, Unidirectionalization of hamiltonian waves, Physics Letters A, vol.126, issue.8-9, pp.501-506, 1988.
DOI : 10.1016/0375-9601(88)90047-3

A. A. Petrov, Variational statement of the problem of liquid motion in a container of finite dimensions, Journal of Applied Mathematics and Mechanics, vol.28, issue.4, pp.917-922, 1964.
DOI : 10.1016/0021-8928(64)90077-2

A. C. Radder, HAMILTONIAN DYNAMICS OF WATER WAVES, Adv. Coast. Ocean Engng, vol.4, issue.4, pp.21-59, 1999.
DOI : 10.1142/9789812797551_0002

R. Salmon, Hamiltonian Fluid Mechanics, Annual Review of Fluid Mechanics, vol.20, issue.1, pp.225-256, 1988.
DOI : 10.1146/annurev.fl.20.010188.001301

F. Serre, Contribution à l'étude des écoulements permanents et variables dans les canaux, pp.374-388, 1953.

J. J. Stoker, Water Waves: The mathematical theory with applications. Interscience, 1957.

C. H. Su and C. S. Gardner, Korteweg???de Vries Equation and Generalizations. III. Derivation of the Korteweg???de Vries Equation and Burgers Equation, Journal of Mathematical Physics, vol.10, issue.3, pp.536-539, 1969.
DOI : 10.1063/1.1664873

C. E. Synolakis, The runup of solitary waves, Journal of Fluid Mechanics, vol.87, issue.-1, pp.523-545, 1987.
DOI : 10.1017/S0022112058000331

C. E. Synolakis and E. N. Bernard, Tsunami science before and beyond Boxing Day 2004, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.125, issue.5743, pp.2231-2265, 2004.
DOI : 10.1126/science.1114576

M. D. Thomas and A. D. Craik, Three-wave resonance for free-surface flows over flexible boundaries, Journal of Fluids and Structures, vol.2, issue.4, pp.323-338, 1988.
DOI : 10.1016/S0889-9746(88)90044-8

K. Trulsen, I. Kliakhandler, K. B. Dysthe, and M. G. Velarde, On weakly nonlinear modulation of waves on deep water, Physics of Fluids, vol.12, issue.10, pp.2432-2437, 2000.
DOI : 10.1063/1.1287856

G. B. Whitham, A general approach to linear and non-linear dispersive waves using a Lagrangian, Journal of Fluid Mechanics, vol.none, issue.02, pp.273-283, 1965.
DOI : 10.1017/S0022112065000745

T. Y. Wu, A unified theory for modeling water waves, Adv. App. Mech, vol.37, issue.8, pp.1-88, 2001.
DOI : 10.1016/S0065-2156(00)80004-6

A. Yahalom and D. Lynden-bell, Simplified variational principles for barotropic magnetohydrodynamics, Journal of Fluid Mechanics, vol.31, issue.5, pp.235-265, 2008.
DOI : 10.1086/164610

V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, Journal of Applied Mechanics and Technical Physics, vol.10, issue.no. 4, pp.190-194, 1968.
DOI : 10.1007/BF00913182

V. E. Zakharov and E. A. Kuznetsov, Hamiltonian formalism for nonlinear waves, Uspekhi Fizicheskih Nauk, vol.167, issue.11, pp.1137-1168, 1997.
DOI : 10.3367/UFNr.0167.199711a.1137

J. G. Zhou, D. M. Causon, D. M. Ingram, and C. G. Mingham, Numerical solutions of the shallow water equations with discontinuous bed topography, International Journal for Numerical Methods in Fluids, vol.33, issue.8, pp.769-788, 2002.
DOI : 10.1002/fld.243