Nonlinear Fourier analysis of deep-water, random surface waves: Theoretical formulation and experimental observations of rogue waves (original) (raw)
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Nonlinear Processes in Geophysics Discussions, 2016
The nonlinear Schrödinger equation (NLS equation) describing weakly rotational wave packets in an infinity-depth fluid in the Lagrangian coordinates is derived. The vorticity is assumed to be an arbitrary function of the Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. It is proved that the modulation instability criteria of the low-vorticity waves and deep water potential waves coincide. All the known solutions of the NLS equation for rogue waves are applicable to the low-vorticity waves. The effect of vorticity is manifested in a shift of the wave number in the carrier wave. In case of vorticity dependence on the vertical Lagrangian coordinate only (the Gouyon waves) this shift is constant. In a more general case, where the vorticity is dependent on both Lagrangian coordinates, the shift of the wave number is horizontally heterogeneous. There is a special case with the Gerstner waves where the vorticity is proportional to the square o...
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Interpretation of random wave field on a shallow water in terms of Fourier spectra is not adequate, when wave amplitudes are not infinitesimally small. A nonlinearity of wave fields leads to the harmonic interactions and random variation of Fourier spectra. As has been shown by Osborne and his co-authors, a more adequate analysis can be performed in terms of nonlinear modes representing cnoidal waves; a spectrum of such modes remains unchanged even in the process of nonlinear mode interactions. Here we show that there is an alternative and more simple analysis of random wave fields on shallow water, which can be presented in terms of interacting Korteweg–de Vries solitons. The data processing of random wave field is developed on the basis of inverse scattering method. The soliton component obscured in a random wave field is determined and a corresponding distribution function of number of solitons on their amplitudes is constructed. The approach developed is illustrated by means of ...
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Physics of Fluids, 2000
We propose a new approach for modeling weakly nonlinear waves, based on enhancing truncated amplitude equations with exact linear dispersion. Our example is based on the nonlinear Schrödinger ͑NLS͒ equation for deep-water waves. The enhanced NLS equation reproduces exactly the conditions for nonlinear four-wave resonance ͑the ''figure 8'' of Phillips͒ even for bandwidths greater than unity. Sideband instability for uniform Stokes waves is limited to finite bandwidths only, and agrees well with exact results of McLean; therefore, sideband instability cannot produce energy leakage to high-wave-number modes for the enhanced equation, as reported previously for the NLS equation. The new equation is extractable from the Zakharov integral equation, and can be regarded as an intermediate between the latter and the NLS equation. Being solvable numerically at no additional cost in comparison with the NLS equation, the new model is physically and numerically attractive for investigation of wave evolution.
Physics Letters A, 2012
We consider the effect of the wind and the dissipation on the nonlinear stages of the modulational instability. By applying a suitable transformation, we map the forced/damped Nonlinear Schrödinger (NLS) equation into the standard NLS with constant coefficients. The transformation is valid as long as |Γt| 1, with Γ the growth/damping rate of the waves due to the wind/dissipation. Approximate rogue wave solutions of the equation are presented and discussed. The results shed some lights on the effects of wind and dissipation on the formation of rogue waves.
Internal solitary waves in the ocean: Analysis using the periodic, inverse scattering transform
Mathematics and Computers in Simulation, 2009
The periodic, inverse scattering transform (PIST) is a powerful analytical tool in the theory of integrable, nonlinear evolution equations. Osborne pioneered the use of the PIST in the analysis of data form inherently nonlinear physical processes. In particular, Osborne's so-called nonlinear Fourier analysis has been successfully used in the study of waves whose dynamics are (to a good approximation) governed by the Korteweg-de Vries equation. In this paper, the mathematical details and a new application of the PIST are discussed. The numerical aspects of and difficulties in obtaining the nonlinear Fourier (i.e., PIST) spectrum of a physical data set are also addressed. In particular, an improved bracketing of the "spectral eigenvalues" (i.e., the ±1 crossings of the Floquet discriminant) and a new root-finding algorithm for computing the latter are proposed. Finally, it is shown how the PIST can be used to gain insightful information about the phenomenon of soliton-induced acoustic resonances, by computing the nonlinear Fourier spectrum of a data set from a simulation of internal solitary wave generation and propagation in the Yellow Sea.
2017
The paper describes a new derivation of the NLS equation, based on a Lagrangian coordinates approach, in the presence of weak vorticity. First, an introduction presents several previously existing derivations of the NLS equation, and offers an interesting review of recent developments designed to take vorticity into account. Then, the Lagrange coordinates, and associated general equations are presented in section 2, while the new NLS equation related to this framework is derived in section 3. Several results are presented at the end of section 3, and in section 4 (only those related to envelope soliton solutions), and summarized in section 5. The paper is relatively well structured, even if several typos remain. Globally, several new results can be found in the manuscript, and for all these reasons, I recommend publication, after some modifi-
Rogue waves: analytical predictions
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2013
Rogue waves observed in the ocean and elsewhere are often modelled by certain solutions of the nonlinear Schrodinger equation, describing the modulational instability of a plane wave and the subsequent development of multi-phase nonlinear wavetrains. In this paper, we describe how integrability and application of the inverse scattering transform can be used to construct a class of explicit asymptotic solutions that describe this process. We discuss the universal mechanism of the onset of multiphase nonlinear waves (rogue waves) through the sequence of successive multi-breather wavetrains. Some applications to ocean waves and laboratory experiments are presented.