M. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM Studies in Applied Mathematics, vol.4, 1981.
DOI : 10.1137/1.9781611970883

P. Aceves-sánchez, A. A. Minzoni, and P. Panayotaros, Numerical study of a nonlocal model for water-waves with variable depth, Wave Motion, vol.50, issue.1, pp.80-93, 2013.
DOI : 10.1016/j.wavemoti.2012.07.002

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

W. Choi and R. Camassa, Exact Evolution Equations for Surface Waves, Journal of Engineering Mechanics, vol.125, issue.7, pp.756-760, 1999.
DOI : 10.1061/(ASCE)0733-9399(1999)125:7(756)

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, P. Guyenne, and H. Kalisch, Hamiltonian long-wave expansions for free surfaces and interfaces, Communications on Pure and Applied Mathematics, vol.9, issue.12, pp.1587-1641, 2005.
DOI : 10.1002/cpa.20098

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. I. Dyachenko, E. A. Kuznetsov, M. D. Spector, and V. E. Zakharov, Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping), Physics Letters A, vol.221, issue.1-2, pp.73-79, 1996.
DOI : 10.1016/0375-9601(96)00417-3

J. De-frutos and J. M. Sanz-serna, An easily implementable fourth-order method for the time integration of wave problems, Journal of Computational Physics, vol.103, issue.1, pp.160-168, 1992.
DOI : 10.1016/0021-9991(92)90331-R

M. Ehrnström, M. D. Groves, and E. Wahlén, On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type, Nonlinearity, vol.25, issue.10, pp.2903-2936, 2012.
DOI : 10.1088/0951-7715/25/10/2903

M. Ehrnström and H. Kalisch, Traveling waves for the Whitham equation, Differential Integral Equations, vol.22, pp.1193-1210, 2009.

M. Ehrnström and H. Kalisch, Global Bifurcation for the Whitham Equation, Mathematical Modelling of Natural Phenomena, vol.8, issue.5, pp.13-30, 2013.
DOI : 10.1051/mmnp/20138502

B. Fornberg and G. B. Whitham, A Numerical and Theoretical Study of Certain Nonlinear Wave Phenomena, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.289, issue.1361, pp.373-404, 1978.
DOI : 10.1098/rsta.1978.0064

J. L. Hammack and H. Segur, The Korteweg-de Vries equation and water waves. Part 2. Comparison with experiments, Journal of Fluid Mechanics, vol.43, issue.02, pp.289-314, 1974.
DOI : 10.1103/PhysRevLett.19.1095

V. M. Hur and M. Johnson, Modulational instability in the Whitham equation of water waves. to appear in Stud, Appl. Math

D. Lannes, The Water Waves Problem, Mathematical Surveys and Monographs, vol.188
DOI : 10.1090/surv/188

URL : https://hal.archives-ouvertes.fr/hal-01101991

D. Lannes and J. Saut, Remarks on the full dispersion Kadomtsev-Petviashvli equation, Kinetic and Related Models, vol.6, issue.4, pp.989-1009, 2013.
DOI : 10.3934/krm.2013.6.989

URL : https://hal.archives-ouvertes.fr/hal-01101990

Y. A. Li, J. M. Hyman, and W. Choi, A Numerical Study of the Exact Evolution Equations for Surface Waves in Water of Finite Depth, Studies in Applied Mathematics, vol.303, issue.3, pp.303-324, 2004.
DOI : 10.1111/j.0022-2526.2004.01534.x

P. Milewski, J. Vanden-broeck, and Z. Wang, Dynamics of steep two-dimensional gravity???capillary solitary waves, Journal of Fluid Mechanics, vol.200, pp.466-477, 2010.
DOI : 10.1017/S0022112095001236

D. Mitsotakis, D. Dutykh, and J. D. Carter, On the nonlinear dynamics of the traveling-wave solutions of the Serre equations, 2014.

D. P. Nicholls and F. Reitich, A new approach to analyticity of Dirichlet-Neumann operators, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, vol.131, issue.06, pp.1411-1433, 2001.
DOI : 10.1017/S0308210500001463

L. Ovsyannikov, To the shallow water theory foundation, Arch. Mech, vol.26, pp.407-422, 1974.

D. H. Peregrine, Calculations of the development of an undular bore, Journal of Fluid Mechanics, vol.224, issue.02, pp.321-330, 1966.
DOI : 10.1007/BF00384031

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

N. Sanford, K. Kodama, J. D. Carter, and H. Kalisch, Stability of traveling wave solutions to the Whitham equation, Physics Letters A, vol.378, issue.30-31, pp.2100-2107, 2014.
DOI : 10.1016/j.physleta.2014.04.067

G. B. Whitham, Variational methods and applications to water waves, Proc. Roy. Soc. London A, pp.6-25, 1967.

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