scholarly journals Transformation of the Peregrine Breather Into Gray Solitons on a Vertically Sheared Current

2021 ◽  
Vol 9 ◽  
Author(s):  
H. C. Hsu ◽  
M. Abid ◽  
Y. Y. Chen ◽  
C. Kharif
Keyword(s):  
Free Surface ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Deep Water ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Research Report ◽  
Nonlinear Schrödinger ◽  
Shear Current ◽  
Peregrine Breather

In this Brief Research Report, we show, within the framework of the nonlinear Schrödinger equation in deep water and in the presence of vorticity (vor-NLS), that the Peregrine breather traveling at the free surface of a shear current of slowly varying vorticity may transform into gray solitons.

scholarly journals Two-dimensional modulation instability of wind waves

2019 ◽  
Vol 5 (4) ◽  
pp. 413-417 ◽  
Author(s):  
Roger Grimshaw
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Water Waves ◽  
Schrodinger Equation ◽  
Modulation Instability ◽  
Wind Waves ◽  
Nonlinear Schrodinger Equation ◽  
Wind Forcing ◽  
Nonlinear Schrödinger ◽  
Peregrine Breather

Abstract It is widely known that deep-water waves are modulationally unstable and that this can be modelled by a nonlinear Schrödinger equation. In this paper, we extend the previous studies of the effect of wind forcing on this instability to water waves in finite depth and in two horizontal space dimensions. The principal finding is that the instability is enhanced and becomes super-exponential and that the domain of instability in the modulation wavenumber space is enlarged. Since the outcome of modulation instability is expected to be the generation of rogue waves, represented within the framework of the nonlinear Schrödinger equation as a Peregrine breather, we also examine the effect of wind forcing on a Peregrine breather. We find that the breather amplitude will grow at twice the rate of a linear instability.

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.

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