Abstract
We investigate and compare soliton turbulence appearing as a result of modulational instability of the homogeneous wave train in three nonlinear models for surface gravity waves: the nonlinear Schrödinger equation, the super compact Zakharov equation, and the fully nonlinear equations written in conformal variables. We show that even at a low level of energy and average wave steepness, the wave dynamics in the nonlinear Schrödinger equation fundamentally differ from the dynamics in more accurate models. We study energy losses of wind waves due to their breaking for large values of total energy in the super compact Zakharov equation and in the exact equations and show that in both models, the wave system loses 50% of energy very slowly, during few days.
Highlights
Turbulence in nonlinear continuous media often accompanied by the appearance of localized nonlinear structures
We investigate and compare soliton turbulence appearing as a result of modulational instability of the homogeneous wave train on the surface of deep water in the framework approximate and exact nonlinear hydrodynamic models
The nonlinear Schrödinger equation (NLS), the super compact Zakharov equation [23] (SCZ), the fully nonlinear equations written in conformal variables [25] (RV)
Summary
Turbulence in nonlinear continuous media often accompanied by the appearance of localized nonlinear structures. We investigate and compare soliton turbulence appearing as a result of modulational instability of the homogeneous wave train on the surface of deep water in the framework approximate and exact nonlinear hydrodynamic models. In 2011 (see papers [20,21]), Dyachenko and Zakharov assuming that all waves propagate in a single direction applied a canonical transformation to remove all cubic nonlinear terms and to drastically simplify fourth-order terms in the truncated Hamiltonian (4) This transformation is possible due to the unexpected cancellation of non-trivial four-wave interactions in the one-dimensional case [22]. We observe soliton turbulence appearing as a result of modulational instability of the homogeneous wave train in the framework nonlinear hydrodynamic models for free surface waves:. The results obtained in the framework of super compact Zakharov equation and the RV equations show that for moderate values of the average slope of the wave for a hundred thousand characteristic periods of the wave, the sea loses about 40–50% of its energy
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