J. Álvarez and A. Durán, Petviashvili type methods for traveling wave computations: I. Analysis of convergence, Journal of Computational and Applied Mathematics, vol.266, issue.8, pp.39-51, 2014.
DOI : 10.1016/j.cam.2014.01.015

K. I. Babenko, Some remarks on the theory of surface waves of finite amplitude, Sov. Math. Dokl, vol.35, issue.5, pp.599-603, 1987.

T. B. Benjamin and J. E. Feir, The disintegration of wavetrains in deep water. Part 1, J. Fluid Mech, vol.27, issue.417, p.19, 1967.

V. I. Bespalov and V. I. Talanov, Filamentary Structure of Light Beams in Nonlinear Liquids, JETP Lett, vol.3, pp.307-326, 1966.

J. P. Boyd, Pseudospectral/Delves???Freeman Computations of the Radiation Coefficient for Weakly Nonlocal Solitary Waves of the Third-Order Nonlinear Schroedinger Equation and Their Relation to Hyperasymptotic Perturbation Theory, Journal of Computational Physics, vol.138, issue.2, pp.665-694, 1997.
DOI : 10.1006/jcph.1997.5840

J. P. Boyd, Radiative decay of weakly nonlocal solitary waves, Wave Motion, vol.27, issue.3, pp.211-221, 1998.
DOI : 10.1016/S0165-2125(97)00043-7

J. P. Boyd, Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics, 1998.
DOI : 10.1007/978-1-4615-5825-5

J. P. Boyd, The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series, Acta Applicandae Mathematicae, vol.56, issue.1, pp.1-98, 1999.
DOI : 10.1023/A:1006145903624

J. P. Boyd, Chebyshev and Fourier Spectral Methods, p.18, 2000.
DOI : 10.1007/978-3-642-83876-7

J. P. Boyd, Why Newton???s method is hard for travelling waves: Small denominators, KAM theory, Arnold???s linear Fourier problem, non-uniqueness, constraints and erratic failure, Mathematics and Computers in Simulation, vol.74, issue.2-3, pp.72-81, 2007.
DOI : 10.1016/j.matcom.2006.10.001

M. A. Branch and A. Grace, MATLAB Optimization Toolbox User's Guide, The Matworks, 1996.

C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, 1988.
DOI : 10.1007/978-3-642-84108-8

C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods Fundamentals in Single Domains. Scientific Computation, 2006.

A. R. Champneys and M. D. Groves, A global investigation of solitary-wave solutions to a two-parameter model for water waves, Journal of Fluid Mechanics, vol.342, pp.199-229, 1997.
DOI : 10.1017/S0022112097005193

A. R. Champneys, J. Vanden-broeck, and G. J. Lord, Do true elevation gravity???capillary solitary waves exist? A numerical investigation, Journal of Fluid Mechanics, vol.454, issue.13, pp.403-417, 2002.
DOI : 10.1017/S0022112001007200

D. Clamond and D. Dutykh, Fast accurate computation of the fully nonlinear solitary surface gravity waves, Computers & Fluids, vol.84, pp.35-38, 2008.
DOI : 10.1016/j.compfluid.2013.05.010
URL : https://hal.archives-ouvertes.fr/hal-00759812

D. Clamond, D. Dutykh, and A. Durán, A plethora of generalised solitary gravity???capillary water waves, Journal of Fluid Mechanics, vol.35, issue.16, pp.664-680, 2015.
DOI : 10.1111/j.0022-2526.2004.01534.x
URL : https://hal.archives-ouvertes.fr/hal-01081798

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

D. Dutykh and D. Clamond, Efficient computation of steady solitary gravity waves, Wave Motion, vol.51, issue.1, pp.86-99, 2014.
DOI : 10.1016/j.wavemoti.2013.06.007
URL : https://hal.archives-ouvertes.fr/hal-00786077

R. Grimshaw, The solitary wave in water of variable depth. Part 2, Journal of Fluid Mechanics, vol.67, issue.03, pp.611-622, 1971.
DOI : 10.1017/S0022112071000739

J. K. Hunter and J. Vanden-broeck, Solitary and periodic gravity???capillary waves of finite amplitude, Journal of Fluid Mechanics, vol.39, issue.-1, pp.205-219, 1983.
DOI : 10.1063/1.862492

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems, Society for Industrial and Applied Mathematics, 1995.
DOI : 10.1137/1.9781611971217

K. Levenberg, A method for the solution of certain non-linear problems in least squares, Quarterly of Applied Mathematics, vol.2, issue.2, pp.164-168, 1944.
DOI : 10.1090/qam/10666

E. Lombardi, Oscillatory integrals and phenomena beyond all algebraic orders with applications to homoclinic orbits in reversible systems, p.13, 2000.

M. L. Lourakis and A. A. Argyros, Is Levenberg-Marquardt the most efficient optimization algorithm for implementing bundle adjustment?, Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1, pp.1526-1531, 2005.
DOI : 10.1109/ICCV.2005.128

K. Madsen, H. Nielsen, and O. Tingleff, Methods for Non-Linear Least Squares Problems, 2004.

D. W. Marquardt, An Algorithm for Least-Squares Estimation of Nonlinear Parameters, Journal of the Society for Industrial and Applied Mathematics, vol.11, issue.2, pp.431-441, 1963.
DOI : 10.1137/0111030

J. W. Mclean, Instabilities of finite-amplitude water waves, Journal of Fluid Mechanics, vol.93, issue.-1, pp.315-330, 1982.
DOI : 10.1017/S0022112075000195

P. Milewski, J. Vanden-broeck, and Z. Wang, Dynamics of steep two-dimensional gravity???capillary solitary waves, Journal of Fluid Mechanics, vol.200, issue.5, pp.466-477, 2010.
DOI : 10.1017/S0022112095001236

J. J. Moré, The Levenberg-Marquardt algorithm: Implementation and theory, Proceedings of the Biennial Conference Held at Dundee, pp.105-116, 1977.
DOI : 10.1137/0111030

H. Nielsen, Damping Parameter in Marquardt's Method, 1999.

J. Nocedal and S. J. Wright, Numerical Optimization, 1999.
DOI : 10.1007/b98874

H. Okamoto and M. Shoji, The Mathematical Theory of Permanent Progressive Water Waves, World Scientific, vol.20, issue.5, 2001.
DOI : 10.1142/4547

L. V. Ovsyannikov, To the shallow water theory foundation, Arch. Mech, vol.26, issue.5, pp.407-422, 1974.

E. I. Parau, J. Vanden-broeck, and M. J. Cooker, Three-dimensional gravity-capillary solitary waves in water of finite depth and related problems, Physics of Fluids, vol.17, issue.12, p.122101, 2005.
DOI : 10.1063/1.2140020

D. E. Pelinovsky and Y. A. Stepanyants, Convergence of Petviashvili's Iteration Method for Numerical Approximation of Stationary Solutions of Nonlinear Wave Equations, SIAM Journal on Numerical Analysis, vol.42, issue.3, pp.1110-1127, 2004.
DOI : 10.1137/S0036142902414232

L. N. Trefethen, Spectral methods in MatLab, Society for Industrial and Applied Mathematics, issue.8, 2000.
DOI : 10.1137/1.9780898719598

J. Vanden-broeck, Gravity-Capillary Free-Surface Flows, p.7, 2010.

J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems, Society for Industrial and Applied Mathematics, 2008.
DOI : 10.1137/1.9780898719680