Journal of Ocean Engineering and Marine Energy

, Volume 3, Issue 4, pp 409–415 | Cite as

Envelope equation for water waves

Soliton turbulence and wavebreaking
  • A. I. Dyachenko
  • D. I. Kachulin
  • V. E. Zakharov
Research Article


Water waves have long been a subject of attention of both mathematicians and physicists. The formulation of the problem is simple enough to be considered fundamental, but as of yet many questions still remain unanswered and many phenomena associated with wind-driven turbulence remain puzzling. We consider a “unidirectional” motion of weakly nonlinear gravity waves, i.e., we assume that the spectrum of the free surface contains only nonnegative wavenumbers. We use remarkably simple form of the water wave equation that we named “the super compact equation”. This new equation includes a nonlinear wave term (à la NLSE) together with an advection term that can describe the initial stage of wave breaking. This equation has also very important property. It allows to introduce exact envelope for waves without assumption of narrowness bandwidth.


Wave breaking Hamiltonian formalism Modulational instability Envelope equation 



This work (except Sect. 5) was supported by Grant “Wave turbulence: theory, numerical simulation, experiment” #14-22-00174 of Russian Science Foundation. Numerical simulation of soliton turbulence (Sect. 5) was supported by the Program of Landau Institute for Theoretical Physics “Dynamics of the Complex Environment”.


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • A. I. Dyachenko
    • 1
    • 2
  • D. I. Kachulin
    • 2
  • V. E. Zakharov
    • 1
    • 2
    • 3
    • 4
  1. 1.Landau Institute for Theoretical PhysicsChernogolovkaRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia
  3. 3.Lebedev Physical Institute RASMoscowRussia
  4. 4.Department of MathematicsUniversity of ArizonaTucsonUSA

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