M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, Society for Industrial & Applied Mathematics, issue.5, 1981.
DOI : 10.1137/1.9781611970883

S. Y. Annenkov and V. I. Shrira, Role of non-resonant interactions in the evolution of nonlinear random water wave fields, Journal of Fluid Mechanics, vol.561, pp.181-207, 2004.
DOI : 10.1017/S0022112006000632

T. B. Benjamin and J. E. Feir, The disintegration of wavetrains in deep water. Part 1, J. Fluid Mech, vol.27, issue.5 6, 1967.

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

J. C. Bronski, M. A. Johnson, and T. Kapitula, An index theorem for the stability of periodic travelling waves of Korteweg???de Vries type, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, vol.141, issue.06, pp.1411141-1173, 2008.
DOI : 10.1017/S0308210510001216

J. R. Cash and A. H. Karp, A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides, ACM Transactions on Mathematical Software, vol.16, issue.3, pp.201-222, 1990.
DOI : 10.1145/79505.79507

P. Denissenko, S. Lukaschuk, and S. Nazarenko, Gravity Wave Turbulence in a Laboratory Flume, Physical Review Letters, vol.99, issue.1, p.14501, 2007.
DOI : 10.1103/PhysRevLett.99.014501

H. R. Dullin, G. A. Gottwald, and D. D. Holm, On asymptotically equivalent shallow water wave equations, Physica D: Nonlinear Phenomena, vol.190, issue.1-2, pp.1-14, 2004.
DOI : 10.1016/j.physd.2003.11.004

D. Dutykh and E. Tobisch, Observation of the inverse energy cascade in the modified Korteweg-de Vries equation, EPL (Europhysics Letters), vol.107, issue.1, p.14001, 2014.
DOI : 10.1209/0295-5075/107/14001

URL : https://hal.archives-ouvertes.fr/hal-00991944

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.

M. Frigo, A fast Fourier transform compiler, Proc. 1999 ACM SIGPLAN Conf. on Programming Language Design and Implementation, pp.169-180, 1999.

M. Frigo and S. G. Johnson, FFTW: an adaptive software architecture for the FFT, Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP '98 (Cat. No.98CH36181), pp.1381-1384, 1998.
DOI : 10.1109/ICASSP.1998.681704

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

R. Grimshaw, INTERNAL SOLITARY WAVES, Environmental Stratified Flows, pp.1-27, 2002.
DOI : 10.1142/9789812797568_0001

URL : https://hal.archives-ouvertes.fr/hal-00302108

R. Grimshaw, D. E. Pelinovsky, E. N. Pelinovsky, and T. Talipova, Wave group dynamics in weakly nonlinear long-wave models, Physica D: Nonlinear Phenomena, vol.159, issue.1-2, pp.35-57, 2001.
DOI : 10.1016/S0167-2789(01)00333-5

R. Grimshaw, E. N. Pelinovsky, T. Talipova, M. S. Ruderman, and R. Erdelyi, Short-Lived Large-Amplitude Pulses in the Nonlinear Long-Wave Model Described by the Modified Korteweg-De Vries Equation, Studies in Applied Mathematics, vol.147, issue.6, pp.189-210, 2005.
DOI : 10.1006/jdeq.1993.1040

E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential- Algebraic Problems, 1996.
DOI : 10.1007/978-3-642-05221-7

M. Haragus and T. Kapitula, On the spectra of periodic waves for infinite-dimensional Hamiltonian systems, Phys. D, vol.237, issue.20, pp.2649-2671, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00469963

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

T. Y. Hou and R. Li, Computing nearly singular solutions using pseudo-spectral methods, Journal of Computational Physics, vol.226, issue.1, pp.379-397, 2007.
DOI : 10.1016/j.jcp.2007.04.014

M. A. Johnson, Stability of Small Periodic Waves in Fractional KdV-Type Equations, SIAM Journal on Mathematical Analysis, vol.45, issue.5, pp.3168-3193, 2005.
DOI : 10.1137/120894397

E. Kartashova, Energy spectra of 2D gravity and capillary waves with narrow frequency band excitation, EPL (Europhysics Letters), vol.97, issue.3, p.30004, 2006.
DOI : 10.1209/0295-5075/97/30004

E. Kartashova, Energy transport in weakly nonlinear wave systems with narrow frequency band excitation, Physical Review E, vol.86, issue.4, p.41129, 2004.
DOI : 10.1103/PhysRevE.86.041129

E. Kartashova, Time scales and structures of wave interaction exemplified with water waves, EPL (Europhysics Letters), vol.102, issue.4, pp.44005-44024, 2004.
DOI : 10.1209/0295-5075/102/44005

A. N. Kolmogorov, The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds Numbers, Proc. R. Soc. Lond. A, pp.9-13, 1890.
DOI : 10.1098/rspa.1991.0075

R. M. Miura, C. S. Gardner, and M. D. , Korteweg???de Vries Equation and Generalizations. II. Existence of Conservation Laws and Constants of Motion, Journal of Mathematical Physics, vol.9, issue.8, p.1204, 1968.
DOI : 10.1063/1.1664701

A. C. Newell and B. Rumpf, Wave Turbulence, Annual Review of Fluid Mechanics, vol.43, issue.1, pp.59-78, 2004.
DOI : 10.1146/annurev-fluid-122109-160807

URL : https://hal.archives-ouvertes.fr/hal-00302012

H. Ono, Algebraic Soliton of the Modified Korteweg-de Vries Equation, Journal of the Physical Society of Japan, vol.41, issue.5, pp.1817-1818, 1976.
DOI : 10.1143/JPSJ.41.1817

M. S. Ruderman, T. Talipova, and E. Pelinovsky, Dynamics of modulationally unstable ionacoustic wavepackets in plasmas with negative ions, J. Plasma Phys, vol.74, issue.05, pp.639-656, 2008.

H. Schamel, A modified Korteweg-de Vries equation for ion acoustic wavess due to resonant electrons, Journal of Plasma Physics, vol.39, issue.03, pp.377-387, 1973.
DOI : 10.1002/cpa.3160180107

E. Tobisch, Energy spectrum of the ensemble of weakly nonlinear gravity-capillary waves on a fluid surface, Journal of Experimental and Theoretical Physics, vol.119, issue.2, pp.359-365, 2014.
DOI : 10.1134/S1063776114080184

L. N. Trefethen, Spectral methods in MatLab, Society for Industrial and Applied Mathematics, issue.7, 2000.
DOI : 10.1137/1.9780898719598

M. P. Tulin and T. Waseda, Laboratory observations of wave group evolution, including breaking effects, Journal of Fluid Mechanics, vol.378, pp.197-232, 1999.
DOI : 10.1017/S0022112098003255

G. B. Whitham, Variational Methods and Applications to Water Waves, Proc. R. Soc. Lond. A, pp.6-25, 1456.

H. Xia, M. Shats, and H. Punzmann, Modulation instability and capillary wave turbulence, EPL (Europhysics Letters), vol.91, issue.1, p.14002, 2006.
DOI : 10.1209/0295-5075/91/14002

URL : http://arxiv.org/abs/1006.4672

N. J. Zabusky and C. C. Galvin, Shallow-water waves, the Korteweg-deVries equation and solitons, Journal of Fluid Mechanics, vol.44, issue.04, pp.811-824, 1971.
DOI : 10.1063/1.1664701

V. E. Zakharov, V. S. Lvov, and G. Falkovich, Kolmogorov Spectra of Turbulence I. Wave Turbulence, Series in Nonlinear Dynamics, 1992.