T. R. Akylas, Higher-order modulation effects on solitary wave envelopes in deep water, Journal of Fluid Mechanics, vol.369, issue.-1, pp.387-397, 1989.
DOI : 10.1017/S0022112083000014

J. D. Carter, Stability and existence of traveling-wave solutions of the two-dimensional nonlinear Schrödinger equation and its higher-order generalizations, 2001.

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

M. Colin and M. Ohta, Stability of solitary waves for derivative nonlinear Schr??dinger equation, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.23, issue.5, pp.753-764, 2006.
DOI : 10.1016/j.anihpc.2005.09.003

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.921-925, 2003.
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.3, pp.404-426, 2011.
DOI : 10.1016/j.euromechflu.2009.10.003

S. J. Hogan, The 4th-order evolution equation for deep-water gravity-capillary waves, Proc. R. Soc. A, pp.359-372, 1823.

S. J. Hogan, The potential form of the fourth-order evolution equation for deep-water gravity???capillary waves, Physics of Fluids, vol.29, issue.10, pp.3479-3480, 1986.
DOI : 10.1063/1.865816

P. A. Janssen, On a fourth-order envelope equation for deep-water waves, Journal of Fluid Mechanics, vol.12, issue.-1, pp.1-11, 1983.
DOI : 10.1017/S0022112062000373

E. Kit and L. Shemer, Spatial versions of the Zakharov and Dysthe evolution equations for deep-water gravity waves, Journal of Fluid Mechanics, vol.450, issue.3, pp.201-205, 2002.
DOI : 10.1017/S0022112001006498

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

E. Lo and C. C. Mei, A numerical study of water-wave modulation based on a higher-order nonlinear Schr??dinger equation, Journal of Fluid Mechanics, vol.22, issue.-1, pp.395-416, 1985.
DOI : 10.1063/1.1694844

E. N. Lorenz, Energy and numerical weather prediction, Tellus, vol.12, issue.2, 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

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

R. Salmon, Practical use of Hamilton's principle, Journal of Fluid Mechanics, vol.117, issue.-1, pp.431-444, 1983.
DOI : 10.1063/1.1706053

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

T. G. Shepherd, Symmetries, Conservation Laws, and Hamiltonian Structure in Geophysical Fluid Dynamics, Adv. Geophys, vol.32, issue.2, pp.287-338, 1990.
DOI : 10.1016/S0065-2687(08)60429-X

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 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

C. Sulem and P. Sulem, The Nonlinear Schrödinger Equation. Self-Focusing and Wave Collapse, 1999.

L. N. Trefethen, Spectral methods in MatLab, Society for Industrial and Applied Mathematics, p.12, 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.3 6, pp.359-373, 1997.
DOI : 10.1017/S0022112097007416

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

J. Wyller, T. Fla, and J. J. Rasmussen, Classification of Kink Type Solutions to the Extended Derivative Nonlinear Schr??dinger Equation, Physica Scripta, vol.57, issue.3, pp.427-435, 1998.
DOI : 10.1088/0031-8949/57/3/015

J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems, SIAM, vol.8, issue.3 10, 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.1990-1994, 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 and A. I. Dyachenko, About shape of giant breather, European Journal of Mechanics - B/Fluids, vol.29, issue.2, pp.127-131, 2010.
DOI : 10.1016/j.euromechflu.2009.10.003

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.8, pp.62-69, 1972.