scholarly journals EVOLUTION OF HIGH-ORDER NONLINEAR PROPERTY USING SPATIAL DEVELOPING NONLINEAR SCHRODINGER EQUATION IN INTERMEDIATE WATER

2020 ◽  
pp. 4
Author(s):  
Hiroaki Kashima ◽  
Nobuhito Mori
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Deep Water ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Property ◽  
Intermediate Water ◽  
Extreme Wave ◽  
Wave Modeling ◽  
Nonlinear Schrödinger

In the last two decades, the extreme wave occurrence, its understanding and prediction in deep water become getting clear. However, there are a few studies about the extreme wave occurrence in intermediate water. The authors reported that the appropriate higher-order nonlinear correction is essential to estimate the extreme wave occurrence using the standard Boussinesq equation. Therefore, the other numerical tool such as nonlinear Schrodinger equation (NLS) will be required from the point of view of the direct extreme wave modeling in intermediate water. The purpose of this study is to investigate the evolution of the high-order nonlinear property in intermediate water using the spatial developing nonlinear Schrodinger equation (SD-NLS) for the deep-water generating extreme wave modeling.

scholarly journals Modified Method of Simplest Equation Applied to the Nonlinear Schrödinger Equation

2018 ◽  
Vol 48 (1) ◽  
pp. 59-68 ◽  
Author(s):  
Nikolay K. Vitanov ◽  
Zlatinka I. Dimitrova
Keyword(s):  
Schrödinger Equation ◽  
Exact Solutions ◽  
Nonlinear Schrödinger Equation ◽  
Deep Water ◽  
Water Waves ◽  
Schrodinger Equation ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Modified Method

AbstractWe consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential equations for deep water waves: the nonlinear Schrödinger equation. It is shown that the methodology works also for other equations of the nonlinear Schrödinger kind.

Note on a modification to the nonlinear Schrödinger equation for application to deep water waves

1979 ◽  
Vol 369 (1736) ◽  
pp. 105-114 ◽  
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Deep Water ◽  
Perturbation Analysis ◽  
Water Waves ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Mean Flow ◽  
Nonlinear Schrödinger ◽  
Deep Water Waves

The ordinary nonlinear Schrödinger equation for deep water waves, found by perturbation analysis to O (∊ 3 ) in the wave-steepness ∊ ═ ka , is shown to compare rather unfavourably with the exact calculations of Longuet-Higgins (1978 b ) for ∊ > 0.15, say. We show that a significant improvement can be achieved by taking the perturbation analysis one step further O (∊ 4 ). The dominant new effect introduced to order ∊ 4 is the mean flow response to non-uniformities in the radiation stress caused by modulation of a finite amplitude wave.

scholarly journals The Lagrange form of the nonlinear Schrödinger equation for low-vorticity waves in deep water: rogue wave aspect

2016 ◽  
Author(s):  
Anatoly Abrashkin ◽  
Efim Pelinovsky
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Deep Water ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Wave Number ◽  
Lagrangian Coordinates ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Lagrangian Solution

Abstract. 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 of the wave amplitude, and the resulting non-linearity disappears, thus making the equations of the dynamics of the Gerstner wave packet linear. It is shown that the NLS solution for weakly rotational waves in the Eulerian variables can be obtained from the Lagrangian solution by the ordinary change of the horizontal coordinates.

scholarly journals Modeling extreme wave heights from laboratory experiments with the nonlinear Schrödinger equation

2013 ◽  
Vol 1 (5) ◽  
pp. 5261-5293 ◽  
Author(s):  
H. D. Zhang ◽  
C. Guedes Soares ◽  
Z. Cherneva ◽  
M. Onorato
Keyword(s):  
Schrödinger Equation ◽  
Numerical Simulations ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Laboratory Experiments ◽  
Nonlinear Schrodinger Equation ◽  
Extreme Wave ◽  
Nonlinear Schrödinger ◽  
Wave Basin ◽  
Wave Heights

Abstract. Spatial variation of nonlinear wave groups with different initial envelope shapes is theoretically studied first, confirming that the simplest nonlinear theoretical model is capable of describing the evolution of propagating wave packets in deep water. Moreover, three groups of laboratory experiments run in the wave basin of CEHIPAR are systematically compared with the numerical simulations of the nonlinear Schrödinger equation. Although a small overestimation is detected, especially in the set of experiments characterized by higher initial wave steepness, the numerical simulations still display a high degree of agreement with the laboratory experiments. Therefore, the nonlinear Schrödinger equation catches the essential characteristics of the extreme waves and provides an important physical insight into their generation. The modulation instability, resulted by the quasi-resonant four wave interaction in a unidirectional sea state, can be indicated by the coefficient of kurtosis, which shows an appreciable correlation with the extreme wave height and hence is used in the modified Edgeworth-Rayleigh distribution. Finally, some statistical properties on the maximum wave heights in different sea states have been related with the initial Benjamin-Feir Index.

Waves in deep water based on the nonlinear Schrödinger equation with variable coefficients

2014 ◽  
Vol 92 (10) ◽  
pp. 1158-1165
Author(s):  
H.I. Abdel-Gawad
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Deep Water ◽  
Water Waves ◽  
Schrodinger Equation ◽  
Variable Coefficients ◽  
Nonlinear Schrodinger Equation ◽  
Time Dependent ◽  
Nonlinear Schrödinger ◽  
First Case

It has been shown that progression of waves in deep water is described by the nonlinear Schrödinger equation with time-dependent diffraction and nonlinearity coefficients. Investigation of the solutions is done here in the two cases when the coefficients are proportional or otherwise. In the first case, it is shown that the water waves are traveling at time-dependent speed and are periodic waves, which are coupled to solitons or elliptic waves seen in the noninertial frames. In the inertial frames wave modulation instability is visualized. In the second case, and when the diffraction coefficient dominates the nonlinearity, water waves collapse with unbounded amplitude at finite time. Exact solutions are found here by using the extended unified method together, while presenting a new algorithm for treating nonlinear coupled partial differential equations.

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