C. J. Amick and K. Kirchgässner, A theory of solitary water-waves in the presence of surface tension, Archive for Rational Mechanics and Analysis, vol.105, issue.1, pp.1-49, 1989.
DOI : 10.1007/BF00251596

D. Antonopoulos, V. Dougalis, and D. Mitsotakis, Error Estimates for Galerkin Approximations of the Serre Equations, SIAM Journal on Numerical Analysis, vol.55, issue.2, p.24, 2017.
DOI : 10.1137/16M1078355

D. Antonopoulos, V. Dougalis, and D. Mitsotakis, On Galerkin-Finite Element Methods for the Camassa-Holm equation, p.12, 2017.

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

J. L. Bona and M. Chen, A Boussinesq system for two-way propagation of nonlinear dispersive waves, Physica D: Nonlinear Phenomena, vol.116, issue.1-2, pp.191-224, 1998.
DOI : 10.1016/S0167-2789(97)00249-2

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

B. Buffoni, M. D. Groves, and J. F. Toland, A Plethora of Solitary Gravity-Capillary Water Waves with Nearly Critical Bond and Froude Numbers, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.354, issue.1707, pp.354575-607, 1707.
DOI : 10.1098/rsta.1996.0020

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Physical Review Letters, vol.337, issue.11, pp.1661-1664, 1993.
DOI : 10.1098/rsta.1991.0133

Y. Y. Chen, C. Kharif, J. H. Yang, H. C. Hsu, J. Touboul et al., An experimental study of steep solitary wave reflection at a vertical wall, European Journal of Mechanics - B/Fluids, vol.49, issue.A, pp.4920-4948, 2015.
DOI : 10.1016/j.euromechflu.2014.07.003
URL : https://hal.archives-ouvertes.fr/hal-01203293

D. Clamond, D. Dutykh, and A. Durán, A plethora of generalised solitary gravity???capillary water waves, Journal of Fluid Mechanics, vol.35, issue.5, pp.664-680, 2015.
DOI : 10.1111/j.0022-2526.2004.01534.x
URL : https://hal.archives-ouvertes.fr/hal-01081798

D. Clamond, D. Dutykh, and A. Galligo, Algebraic method for constructing singular steady solitary waves: a case study, Proc. R. Soc. Lond. A, pp.472-2016, 2191.
DOI : 10.1088/0951-7715/22/1/R01
URL : https://hal.archives-ouvertes.fr/hal-01290471

D. Clamond, D. Dutykh, and D. Mitsotakis, Conservative modified Serre???Green???Naghdi equations with improved dispersion characteristics, Communications in Nonlinear Science and Numerical Simulation, vol.45, issue.6, pp.245-257, 2017.
DOI : 10.1016/j.cnsns.2016.10.009
URL : https://hal.archives-ouvertes.fr/hal-01232370

W. Craig, P. Guyenne, J. Hammack, D. Henderson, and C. Sulem, Solitary water wave interactions, Physics of Fluids, vol.8, issue.5, pp.57106-57130, 2006.
DOI : 10.1137/S1111111102411298

P. Daripa and R. K. Dash, A class of model equations for bi-directional propagation of capillary???gravity waves, International Journal of Engineering Science, vol.41, issue.2, pp.201-218, 2003.
DOI : 10.1016/S0020-7225(02)00180-5

R. K. Dash and P. Daripa, Analytical and numerical studies of a singularly perturbed Boussinesq equation, Applied Mathematics and Computation, vol.126, issue.1, pp.1-30, 2002.
DOI : 10.1016/S0096-3003(01)00166-7

F. Dias and C. Kharif, NONLINEAR GRAVITY AND CAPILLARY-GRAVITY WAVES, Annual Review of Fluid Mechanics, vol.31, issue.1, pp.301-346, 1999.
DOI : 10.1146/annurev.fluid.31.1.301

F. Dias, D. Menasce, and J. Vanden-broeck, Numerical study of capillary-gravity solitary waves, Eur. J. Mech. B/Fluids, vol.15, issue.1 8, pp.17-36, 1996.

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
URL : http://opus.bath.ac.uk/26674/1/PLAgSerreMilewskiDiasResub.pdf

E. Dinvay, D. Moldabayev, D. Dutykh, and H. Kalisch, The Whitham equation with surface tension, Nonlinear Dynamics, vol.9, issue.2, pp.1125-1138, 2005.
DOI : 10.1007/BF00913182
URL : https://hal.archives-ouvertes.fr/hal-01504047

V. A. Dougalis, A. Durán, M. A. Lopez-marcos, and D. E. Mitsotakis, A Numerical Study of the Stability of Solitary Waves of??the??Bona???Smith Family of Boussinesq Systems, Journal of Nonlinear Science, vol.15, issue.6, pp.595-607, 2007.
DOI : 10.1090/conm/200/02517

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

D. Dutykh, D. Clamond, and A. Durán, Efficient computation of capillary???gravity generalised solitary waves, Wave Motion, vol.65, issue.8, pp.1-16
DOI : 10.1016/j.wavemoti.2016.04.007

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

E. Falcon, C. Laroche, and S. Fauve, Observation of Depression Solitary Surface Waves on a Thin Fluid Layer, Physical Review Letters, vol.85, issue.20, p.204501, 2002.
DOI : 10.1017/S0022112078000713

W. H. Hager, Wilfrid Noel Bond and the Bond number, Journal of Hydraulic Research, vol.6, issue.23, pp.3-9, 2007.
DOI : 10.1080/14786443708565150

M. Haragus, Model equations for water waves in the presence of surface tension, Eur. J. Mech. B/Fluids, vol.15, issue.5, pp.471-492, 1996.

M. Haragus-courcelle and A. Il-'ichev, Three-dimensional solitary waves in the presence of additional surface effects, European Journal of Mechanics - B/Fluids, vol.17, issue.5, pp.739-768, 1998.
DOI : 10.1016/S0997-7546(98)80023-X

J. K. Hunter and J. Scheurle, Existence of perturbed solitary wave solutions to a model equation for water waves, Physica D: Nonlinear Phenomena, vol.32, issue.2, pp.253-268, 1988.
DOI : 10.1016/0167-2789(88)90054-1

V. M. Hur and M. A. Johnson, Modulational instability in the Whitham equation with surface tension and vorticity. Nonlinear Analysis: Theory, Methods & Applications, vol.129, pp.104-118, 2005.

T. Kawahara, Oscillatory Solitary Waves in Dispersive Media, Journal of the Physical Society of Japan, vol.33, issue.1, pp.260-264, 1972.
DOI : 10.1143/JPSJ.33.260

D. Lannes, The water waves problem: Mathematical analysis and asymptotics, 2013.
DOI : 10.1090/surv/188
URL : https://hal.archives-ouvertes.fr/hal-01101991

P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Communications on Pure and Applied Mathematics, vol.15, issue.5, pp.467-490, 1968.
DOI : 10.1080/14786449508620739

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 : 10.1080/14786449508620739

S. Liao, Do peaked solitary water waves indeed exist?, Communications in Nonlinear Science and Numerical Simulation, vol.19, issue.6, pp.1792-1821, 2014.
DOI : 10.1016/j.cnsns.2013.09.042
URL : http://arxiv.org/pdf/1204.3354.pdf

N. K. Lowman, M. A. Hoefer, and G. A. , Interactions of large amplitude solitary waves in viscous fluid conduits, Journal of Fluid Mechanics, vol.2, issue.21, pp.372-384, 2014.
DOI : 10.1029/GL011i011p01161

Y. Matsuno, Hamiltonian formulation of the extended Green???Naghdi equations, Physica D: Nonlinear Phenomena, vol.301, issue.302, pp.1-7, 2005.
DOI : 10.1016/j.physd.2015.03.001

J. W. Miles, Obliquely interacting solitary waves, Journal of Fluid Mechanics, vol.9, issue.01, pp.157-169, 1920.
DOI : 10.1143/JPSJ.34.1093

S. M. Mirie and C. H. Su, Collisions between two solitary waves. Part 2. A numerical study, Journal of Fluid Mechanics, vol.95, issue.-1, pp.475-492, 1982.
DOI : 10.1017/S0022112071002295

D. Mitsotakis, D. Dutykh, A. Assylbekuly, and D. Zhakebayev, On weakly singular and fully nonlinear travelling shallow capillary???gravity waves in the critical regime, Physics Letters A, vol.381, issue.20, pp.3811719-1726, 2006.
DOI : 10.1016/j.physleta.2017.03.041
URL : https://hal.archives-ouvertes.fr/hal-01388481

D. Mitsotakis, B. Ilan, and D. Dutykh, On the Galerkin/Finite-Element Method for the Serre Equations, Journal of Scientific Computing, vol.73, issue.6, pp.166-195, 2014.
DOI : 10.1016/j.coastaleng.2012.09.005
URL : https://hal.archives-ouvertes.fr/hal-00834064

T. G. Myers, Thin Films with High Surface Tension, SIAM Review, vol.40, issue.3, pp.441-462, 1998.
DOI : 10.1137/S003614459529284X
URL : http://www.maths.bath.ac.uk/~masjde/MSc/CaseStudies/CaseStudy_contactlens/CL3.pdf

B. A. Packham, Capillary???gravity waves against a vertical cliff, Mathematical Proceedings of the Cambridge Philosophical Society, vol.232, issue.03, pp.827-832, 1968.
DOI : 10.1090/S0002-9904-1946-08643-7

F. Serre, Contribution à l'étude des écoulements permanents et variables dans les canaux, pp.830-872, 1953.

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

P. Sprenger and M. A. Hoefer, Shock Waves in Dispersive Hydrodynamics with Nonconvex Dispersion, SIAM Journal on Applied Mathematics, vol.77, issue.1, pp.26-50, 2008.
DOI : 10.1137/16M1082196

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 : 10.1103/PhysRevLett.17.996

C. H. Su and R. M. Mirie, On head-on collisions between two solitary waves, Journal of Fluid Mechanics, vol.46, issue.03, pp.509-525, 1980.
DOI : 10.1103/PhysRevLett.19.1095
URL : http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA067856&Location=U2&doc=GetTRDoc.pdf

G. B. Whitham, Linear and nonlinear waves, p.73000, 1999.

F. Chambéry and L. , Campus Scientifique, F-73376 Le Bourget-du-Lac Cedex, France E-mail address: Denys.Dutykh@univ-smb

M. Hoefer, E-mail address: hoefer@colorado