M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, The Inverse Scattering Transform-Fourier Analysis for Nonlinear Problems, Studies in Applied Mathematics, vol.34, issue.12, pp.249-315, 1974.
DOI : 10.1002/sapm1974534249

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, Society for Industrial & Applied Mathematics, issue.2, 1981.
DOI : 10.1137/1.9781611970883

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

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

D. Clamond and J. Grue, A fast method for fully nonlinear water-wave computations, Journal of Fluid Mechanics, vol.447, issue.10, pp.337-355, 2001.
DOI : 10.1017/S0022112001006000

P. G. Drazin and R. S. Johnson, Solitons: An introduction, 1989.
DOI : 10.1017/CBO9781139172059

A. I. Dyachenko and V. Zakharov, Is free-surface hydrodynamics an integrable system?, Physics Letters A, vol.190, issue.2, pp.144-148, 1994.
DOI : 10.1016/0375-9601(94)90067-1

A. I. Dyachenko and V. Zakharov, Toward an integrable model of deep water, Physics Letters A, vol.221, issue.1-2, pp.80-84, 1996.
DOI : 10.1016/0375-9601(96)00394-5

A. I. Dyachenko and V. E. Zakharov, Compact equation for gravity waves on deep water, JETP Letters, vol.93, issue.12, pp.701-705, 2007.
DOI : 10.1134/S0021364011120058

A. I. Dyachenko, V. E. Zakharov, and D. I. Kachulin, Collision of two breathers at surface of deep water, pp.15-28, 2012.

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.

F. Fedele and D. Dutykh, Hamiltonian form and solitary waves of the spatial Dysthe equations, JETP Letters, vol.94, issue.12, pp.840-844, 2011.
DOI : 10.1134/S0021364011240039
URL : https://hal.archives-ouvertes.fr/hal-00633389

M. Frigo and S. G. Johnson, The Design and Implementation of FFTW3, Proceedings of the IEEE, pp.216-231, 2005.
DOI : 10.1109/JPROC.2004.840301

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

O. Gramstad and K. Trulsen, Hamiltonian form of the modified nonlinear Schr??dinger equation for gravity waves on arbitrary depth, Journal of Fluid Mechanics, vol.48, issue.2, pp.404-426, 2011.
DOI : 10.1016/j.euromechflu.2009.10.003

S. J. Hogan, The Fourth-Order Evolution Equation for Deep-Water Gravity-Capillary Waves, Proc. R. Soc. A, pp.359-372, 1823.
DOI : 10.1098/rspa.1985.0122

V. P. Krasitskii, On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves, Journal of Fluid Mechanics, vol.71, issue.-1, pp.1-20, 1994.
DOI : 10.1017/S0022112085002488

T. I. Lakoba and J. Yang, A generalized Petviashvili iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity, Journal of Computational Physics, vol.226, issue.2, pp.1668-1692, 2007.
DOI : 10.1016/j.jcp.2007.06.009

M. S. Longuet-higgins and J. H. Fox, Asymptotic theory for the almost-highest solitary wave, Journal of Fluid Mechanics, vol.302, issue.-1, pp.1-19, 1996.
DOI : 10.1137/0128059

M. S. Longuet-higgins and M. J. Fox, Theory of the almost-highest wave: the inner solution, Journal of Fluid Mechanics, vol.76, issue.04, pp.721-741, 1977.
DOI : 10.1007/BF02109151

M. S. Longuet-higgins and M. J. Fox, Theory of the almost-highest wave. Part 2. Matching and analytic extension, Journal of Fluid Mechanics, vol.286, issue.04, pp.769-786, 1978.
DOI : 10.1007/BF02109151

E. N. Lorenz, Energy and numerical weather prediction, Tellus, vol.12, issue.4, pp.364-373, 1960.
DOI : 10.1111/j.2153-3490.1960.tb01323.x

P. Milewski and E. Tabak, A PseudoSpectral Procedure for the Solution of Nonlinear Wave Equations with Examples from Free-Surface Flows, SIAM Journal on Scientific Computing, vol.21, issue.3, pp.1102-1114, 1999.
DOI : 10.1137/S1064827597321532

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

D. 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

V. I. Petviashvili, Equation of an extraordinary soliton, Sov. J. Plasma Phys, vol.2, issue.3, pp.469-472, 1976.

R. L. Seliger and G. B. Whitham, Variational principle in continuous mechanics, Proc. R. Soc. Lond. A, pp.1-25, 1968.

G. Söderlind, Digital filters in adaptive time-stepping, ACM Transactions on Mathematical Software, vol.29, issue.1, pp.1-26, 2003.
DOI : 10.1145/641876.641877

G. Söderlind and L. Wang, Adaptive time-stepping and computational stability, Journal of Computational and Applied Mathematics, vol.185, issue.2, pp.225-243, 2006.
DOI : 10.1016/j.cam.2005.03.008

M. Stiassnie, Note on the modified nonlinear Schr??dinger equation for deep water waves, Wave Motion, vol.6, issue.4, pp.431-433, 1984.
DOI : 10.1016/0165-2125(84)90043-X

M. Stiassnie and L. Shemer, On modifications of the Zakharov equation for surface gravity waves, Journal of Fluid Mechanics, vol.126, issue.-1, pp.47-67, 1984.
DOI : 10.1017/S0022112062000373

L. N. Trefethen, Spectral methods in MatLab, Society for Industrial and Applied Mathematics, vol.6, p.10, 2000.
DOI : 10.1137/1.9780898719598

K. Trulsen and K. B. Dysthe, Frequency downshift in three-dimensional wave trains in a deep basin, Journal of Fluid Mechanics, vol.352, issue.4, pp.359-373, 1997.
DOI : 10.1017/S0022112097007416

N. G. Vakhitov and A. A. Kolokolov, Stationary solutions of the wave equation in a medium with nonlinearity saturation, Radiophysics and Quantum Electronics, vol.3, issue.No. 4, pp.783-789, 1973.
DOI : 10.1007/BF01031343

J. H. Verner, Explicit Runge???Kutta Methods with Estimates of the Local Truncation Error, SIAM Journal on Numerical Analysis, vol.15, issue.4, pp.772-790, 1978.
DOI : 10.1137/0715051

G. B. Whitham, Linear and nonlinear waves, 1999.
DOI : 10.1002/9781118032954

J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems, 2010.
DOI : 10.1137/1.9780898719680

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, Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid, European Journal of Mechanics - B/Fluids, vol.18, issue.3, pp.327-344, 1999.
DOI : 10.1016/S0997-7546(99)80031-4

V. E. Zakharov, P. Guyenne, A. N. Pushkarev, and F. Dias, Wave turbulence in one-dimensional models, Physica D: Nonlinear Phenomena, vol.152, issue.153, pp.573-619, 2001.
DOI : 10.1016/S0167-2789(01)00194-4

V. E. Zakharov and A. B. Shabat, Exact Theory of Two-dimensional Self-focusing and Onedimensional Self-modulation of Waves in Nonlinear Media, Soviet Physics-JETP, vol.34, issue.2, pp.62-69, 1972.