Revista de la
Unión Matemática Argentina
Normal form transformations for modulated deep-water gravity waves
Philippe Guyenne, Adilbek Kairzhan, and Catherine Sulem
Volume 64, no. 2 (2023), pp. 281–308

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Modulation theory is a well-known tool to describe the long-time evolution and stability of small-amplitude, oscillating solutions to dispersive nonlinear partial differential equations. There have been a number of approaches to deriving envelope equations for weakly nonlinear waves. Here we review a systematic method based on Hamiltonian transformation theory and averaging Hamiltonians. In the context of the modulation of two- or three-dimensional deep-water surface waves, this approach leads to a Dysthe equation that preserves the Hamiltonian character of the water wave problem. An explicit calculation of the third-order Birkhoff normal form that eliminates all non-resonant cubic terms yields a non-perturbative procedure for the reconstruction of the free surface. We also present new numerical simulations of this weakly nonlinear approximation using the version with exact linear dispersion. We compare them against computations from the full water wave system and find very good agreement.


  1. M. J. Ablowitz and H. Segur, On the evolution of packets of water waves, J. Fluid Mech. 92 (1979), no. 4, 691–715. MR 0544892.
  2. M. Berti, R. Feola, and F. Pusateri, Birkhoff normal form and long time existence for periodic gravity water waves, Comm. Pure Appl. Math., early view, 2022.
  3. M. Berti, R. Feola, and F. Pusateri, Birkhoff normal form for gravity water waves, Water Waves 3 (2021), no. 1, 117–126. MR 4246390.
  4. U. Brinch-Nielsen and I. G. Jonsson, Fourth order evolution equations and stability analysis for Stokes waves on arbitrary water depth, Wave Motion 8 (1986), no. 5, 455–472. MR 0858522.
  5. H. Chihara, Third order semilinear dispersive equations related to deep water waves, arXiv:math/0404005 [math.AP], 2004.
  6. R. R. Coifman and Y. Meyer, Nonlinear harmonic analysis and analytic dependence, in Pseudodifferential Operators and Applications (Notre Dame, Ind., 1984), 71–78, Proc. Sympos. Pure Math., 43, Amer. Math. Soc., Providence, RI, 1985. MR 0812284.
  7. W. Craig, P. Guyenne, and H. Kalisch, Hamiltonian long-wave expansions for free surfaces and interfaces, Comm. Pure Appl. Math. 58 (2005), no. 12, 1587–1641. MR 2177163.
  8. W. Craig, P. Guyenne, D. P. Nicholls, and C. Sulem, Hamiltonian long-wave expansions for water waves over a rough bottom, Proc. R. Soc. Lond. Ser. A 461 (2005), no. 2055, 839–873. MR 2121939.
  9. W. Craig, P. Guyenne, and C. Sulem, A Hamiltonian approach to nonlinear modulation of surface water waves, Wave Motion 47 (2010), no. 8, 552–563. MR 2734546.
  10. W. Craig, P. Guyenne, and C. Sulem, Normal form transformations and Dysthe's equation for the nonlinear modulation of deep-water gravity waves, Water Waves 3 (2021), no. 1, 127–152. MR 4246391.
  11. W. Craig, U. Schanz, and C. Sulem, The modulational regime of three-dimensional water waves and the Davey-Stewartson system, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), no. 5, 615–667. MR 1470784.
  12. W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys. 108 (1993), no. 1, 73–83. MR 1239970.
  13. W. Craig and C. Sulem, Mapping properties of normal forms transformations for water waves, Boll. Unione Mat. Ital. 9 (2016), no. 2, 289–318. MR 3502161.
  14. W. Craig, C. Sulem, and P.-L. Sulem, Nonlinear modulation of gravity waves: a rigorous approach, Nonlinearity 5 (1992), no. 2, 497–522. MR 1158383.
  15. W. Craig and P. A. Worfolk, An integrable normal form for water waves in infinite depth, Phys. D 84 (1995), no. 3-4, 513–531. MR 1336546.
  16. A. Davey and K. Stewartson, On three-dimensional packets of surface waves, Proc. R. Soc. Lond. Ser. A 338 (1974), 101–110. MR 0349126.
  17. A. I. Dyachenko, D. I. Kachulin, and V. E. Zakharov, Super compact equation for water waves, J. Fluid Mech. 828 (2017), 661–679. MR 3707778.
  18. A. I. Dyachenko and V. E. Zakharov, Is free-surface hydrodynamics an integrable system?, Phys. Lett. A 190 (1994), no. 2, 144–148. MR 1283779.
  19. A. I. Dyachenko and V. E. Zakharov, Compact equation for gravity waves on deep water, JETP Lett. 93 (2011), 701–705.
  20. K. B. Dysthe, Note on a modification to the nonlinear Schrödinger equation for application to deep water waves, Proc. R. Soc. Lond. Ser. A 369 (1979), 105–114.
  21. F. Fedele and D. S. Dutykh, Hamiltonian form and solitary waves of the spatial Dysthe equations, Pis'ma v Zh. Èksper. Teoret. Fiz. 94 (2011), no. 12, 921–925; JETP Lett. 94 (2011), no. 12, 840–844.
  22. O. Gramstad and K. Trulsen, Hamiltonian form of the modified nonlinear Schrödinger equation for gravity waves on arbitrary depth, J. Fluid Mech. 670 (2011), 404–426. MR 2773687.
  23. R. Grande, K. M. Kurianski, and G. Staffilani, On the nonlinear Dysthe equation, Nonlinear Anal. 207 (2021), Paper No. 112292, 36 pp. MR 4220762.
  24. P. Guyenne, A. Kairzhan, and C. Sulem, Hamiltonian Dysthe equation for three-dimensional deep-water gravity waves, Multiscale Model. Simul. 20 (2022), no. 1, 349–378. MR 4395161.
  25. P. Guyenne, A. Kairzhan, C. Sulem, and B. Xu, Spatial form of a Hamiltonian Dysthe equation for deep-water gravity waves, Fluids 6 (2021), no. 3, 103.
  26. P. Guyenne and D. P. Nicholls, A high-order spectral method for nonlinear water waves over moving bottom topography, SIAM J. Sci. Comput. 30 (2007/08), no. 1, 81–101. MR 2377432.
  27. H. Hasimoto and H. Ono, Nonlinear modulation of gravity waves, J. Phys. Soc. Japan 33 (1972), 805–811.
  28. S. J. Hogan, The fourth-order evolution equation for deep-water gravity-capillary waves, Proc. R. Soc. Lond. Ser. A 402 (1985), 359–372.
  29. H. Koch and J.-C. Saut, Local smoothing and local solvability for third order dispersive equations, SIAM J. Math. Anal. 38 (2006/07), no. 5, 1528–1541. MR 2286018.
  30. V. P. Krasitskii, On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves, J. Fluid Mech. 272 (1994), 1–20. MR 1289106.
  31. E. Lo and C. C. Mei, A numerical study of water-wave modulation based on a higher-order nonlinear Schrödinger equation, J. Fluid Mech. 150 (1985), 395–416.
  32. R. Mosincat, D. Pilod, and J.-C. Saut, Global well-posedness and scattering for the Dysthe equation in $L^2(\mathbb{R}^2)$, J. Math. Pures Appl. (9) 149 (2021), 73–97. MR 4238997.
  33. S. Nazarenko, Wave Turbulence, Lecture Notes in Physics, 825, Springer, Heidelberg, 2011. MR 3014432.
  34. L. Shemer, E. Kit, and H.-Y. Jiao, An experimental and numerical study of the spatial evolution of unidirectional nonlinear water-wave groups. Phys. Fluids 14 (2002), 3380–3390.
  35. C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Applied Mathematical Sciences, 139, Springer-Verlag, New York, 1999. MR 1696311.
  36. K. Trulsen, Weakly nonlinear sea surface waves—freak waves and deterministic forecasting, in Geometric Modelling, Numerical Simulation, and Optimization: Applied Mathematics at SINTEF, 191–209, Springer, Berlin, 2007. MR 2348923.
  37. K. Trulsen and K. B. Dysthe, A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water, Wave Motion 24 (1996), no. 3, 281–289. MR 1419980.
  38. K. Trulsen, I. Kliakhandler, K. B. Dysthe, and M. G. Velarde, On weakly nonlinear modulation of waves on deep water, Phys. Fluids 12 (2000), no. 10, 2432–2437. MR 1789997.
  39. S. Wu, The quartic integrability and long-time existence of steep water waves in 2D, arXiv:2010.09117v2 [math.AP], 2021.
  40. L. Xu and P. Guyenne, Numerical simulation of three-dimensional nonlinear water waves, J. Comput. Phys. 228 (2009), no. 22, 8446–8466. MR 2574099.
  41. V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys. 9 (1968), 190–194.
  42. V. Zakharov, Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid, Eur. J. Mech. B Fluids 18 (1999), no. 3, 327–344. MR 1701696.