D. C. Antonopoulos and V. A. Dougalis, Error estimates for Galerkin approximations of the ???classical??? Boussinesq system, Mathematics of Computation, vol.82, issue.282, pp.689-717, 2013.
DOI : 10.1090/S0025-5718-2012-02663-9

D. C. Antonopoulos, V. A. Dougalis, and D. E. Mitsotakis, Galerkin approximations of the periodic solutions of Boussinesq systems, Bulletin of Greek Math. Soc, vol.57, pp.13-30, 2010.

D. C. Antonopoulos, V. A. Dougalis, and D. E. Mitsotakis, Numerical solution of Boussinesq systems of the Bona???Smith family, Applied Numerical Mathematics, vol.60, issue.4, pp.314-336, 2010.
DOI : 10.1016/j.apnum.2009.03.002

URL : https://hal.archives-ouvertes.fr/hal-00371943

D. C. Antonopoulos and V. D. Dougalis, Numerical solution of the ???classical??? Boussinesq system, Mathematics and Computers in Simulation, vol.82, issue.6, pp.984-1007, 2012.
DOI : 10.1016/j.matcom.2011.09.006

E. Barthélémy, Nonlinear shallow water theories for coastal waves, pp.315-337, 2004.

T. B. Benjamin and J. Lighthill, On Cnoidal Waves and Bores, Proc. R. Soc. London, pp.448-460, 1954.
DOI : 10.1098/rspa.1954.0172

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, V. A. Dougalis, O. A. Karakashian, and W. R. Mckinney, Conservative, High-Order Numerical Schemes for the Generalized Korteweg-de Vries Equation, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.351, issue.1695, pp.107-164, 1995.
DOI : 10.1098/rsta.1995.0027

P. Bonneton, F. Chazel, D. Lannes, F. Marche, and M. Tissier, A splitting approach for the fully nonlinear and weakly dispersive Green???Naghdi model, Journal of Computational Physics, vol.230, issue.4, pp.1479-1498, 2011.
DOI : 10.1016/j.jcp.2010.11.015

URL : https://hal.archives-ouvertes.fr/hal-00482564

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, vol.17, pp.55-108

J. D. Carter and R. Cienfuegos, The kinematics and stability of solitary and cnoidal wave solutions of the Serre equations, European Journal of Mechanics - B/Fluids, vol.30, issue.3, pp.259-268, 2011.
DOI : 10.1016/j.euromechflu.2010.12.002

F. Chazel, D. Lannes, and F. Marche, Numerical Simulation of Strongly Nonlinear and Dispersive Waves Using a Green???Naghdi Model, Journal of Scientific Computing, vol.37, issue.3, pp.105-116, 2011.
DOI : 10.1007/s10915-010-9395-9

URL : https://hal.archives-ouvertes.fr/hal-00482561

H. Chen, M. Chen, and N. Nguyen, Cnoidal wave solutions to Boussinesq systems, Nonlinearity, vol.20, issue.6, pp.1443-1461, 2007.
DOI : 10.1088/0951-7715/20/6/007

M. Chen, Solitary-wave and multi-pulsed traveling-wave solutions of boussinesq systems, Applicable Analysis, vol.19, issue.4, pp.213-240, 2000.
DOI : 10.1080/00036810008840844

W. Choi and R. Camassa, Exact Evolution Equations for Surface Waves, Journal of Engineering Mechanics, vol.125, issue.7, p.756, 1999.
DOI : 10.1061/(ASCE)0733-9399(1999)125:7(756)

W. Craig, P. Guyenne, J. Hammack, D. Henderson, and C. Sulem, Solitary water wave interactions, Physics of Fluids, vol.18, issue.5, p.57106, 2006.
DOI : 10.1063/1.2205916

F. Dias and P. Milewski, On the fully-nonlinear shallow-water generalized Serre equations, Physics Letters A, vol.374, issue.8, pp.1049-1053, 2010.
DOI : 10.1016/j.physleta.2009.12.043

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

A. Duran, D. Dutykh, and D. Mitsotakis, On the Gallilean invariance of some nonlinear dispersive wave equations, Stud. Appl. Math, 2013.

D. Dutykh, D. Clamond, P. Milewski, and D. Mitsotakis, Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations, European Journal of Applied Mathematics, vol.9, issue.05, pp.761-787, 2013.
DOI : 10.1017/S0022112065000745

URL : https://hal.archives-ouvertes.fr/hal-00587994

D. Dutykh, T. Katsaounis, and D. Mitsotakis, Finite volume schemes for dispersive wave propagation and runup, Journal of Computational Physics, vol.230, issue.8, pp.3035-3061, 2011.
DOI : 10.1016/j.jcp.2011.01.003

URL : https://hal.archives-ouvertes.fr/hal-00472431

G. A. El, R. H. Grimshaw, and N. F. Smyth, Unsteady undular bores in fully nonlinear shallow-water theory, Physics of Fluids, vol.18, issue.2, p.27104, 2006.
DOI : 10.1063/1.2175152

G. A. El, R. H. Grimshaw, and N. F. Smyth, Asymptotic description of solitary wave trains in fully nonlinear shallow-water theory, Physica D: Nonlinear Phenomena, vol.237, issue.19, pp.2372423-2435, 2008.
DOI : 10.1016/j.physd.2008.03.031

J. G. Esler and J. D. Pearce, Dispersive dam-break and lock-exchange flows in a two-layer fluid, Journal of Fluid Mechanics, vol.667, pp.555-585, 2011.
DOI : 10.1017/S0022112010004593

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

A. V. Gurevich and L. P. Pitaevskii, Nonstationary structure of a collisionless shock wave, Sov. Phys. -JETP, vol.38, pp.291-297, 1974.

S. Israwi, Large time existence for 1D Green-Naghdi equations. Nonlinear Analysis: Theory, Methods & Applications, vol.74, issue.1, pp.81-93, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00415875

R. S. Johnson, Camassa???Holm, Korteweg???de Vries and related models for water waves, Journal of Fluid Mechanics, vol.455, pp.63-82, 2002.
DOI : 10.1017/S0022112001007224

D. Lannes and P. Bonneton, Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation, Physics of Fluids, vol.21, issue.1, p.16601, 2009.
DOI : 10.1063/1.3053183

O. , L. Métayer, S. Gavrilyuk, and S. Hank, A numerical scheme for the Green-Naghdi model, J. Comp. Phys, vol.229, issue.6, pp.2034-2045, 2010.

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. W. Lord-rayleigh, Waves, Phil. Mag, vol.1, pp.257-279, 1876.
DOI : 10.1007/978-3-540-45626-1_8

D. H. Peregrine, Long waves on a beach, Journal of Fluid Mechanics, vol.13, issue.04, pp.815-827, 1967.
DOI : 10.1017/S0022112067002605

M. H. Schultz, Spline Analysis, 1973.

F. J. Seabra-santos, D. P. Renouard, and A. M. Temperville, Numerical and experimental study of the transformation of a solitary wave over a shelf or isolated obstacle, Journal of Fluid Mechanics, vol.274, issue.-1, pp.117-134, 1987.
DOI : 10.1017/S0022112069002461

F. Serre, ContributionàContributionà l'´ etude desécoulementsdesécoulements permanents et variables dans les canaux, pp.374-388, 1953.

F. Serre, ContributionàContributionà l'´ etude desécoulementsdesécoulements permanents et variables dans les canaux, pp.830-872, 1953.

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

G. Wei, J. T. Kirby, S. T. Grilli, and R. Subramanya, A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves, Journal of Fluid Mechanics, vol.107, issue.-1, pp.71-92, 1995.
DOI : 10.1063/1.865459

G. B. Whitham, Non-linear dispersion of water waves, Journal of Fluid Mechanics, vol.25, issue.02, pp.399-412, 1967.
DOI : 10.1017/S0022112067000424

G. B. Whitham, Linear and nonlinear waves, 1999.

Y. Zhang, A. B. Kennedy, N. Panda, C. Dawson, and J. J. Westerink, Boussinesq???Green???Naghdi rotational water wave theory, Coastal Engineering, vol.73, pp.13-27, 2013.
DOI : 10.1016/j.coastaleng.2012.09.005