Validity and Non-Validity of the Nonlinear Schrödinger Equation as a Model for Water Waves

2016 ◽  
pp. 121-139 ◽  
Author(s):  
Guido Schneider
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Water Waves ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger

scholarly journals The Effect of Random Wind Forcing in the Nonlinear Schrödinger Equation

Fluids ◽  
2019 ◽  
Vol 4 (3) ◽  
pp. 121
Author(s):  
Leo Dostal
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Solitary Waves ◽  
Water Waves ◽  
Schrodinger Equation ◽  
Multiple Scales ◽  
Ocean Waves ◽  
Nonlinear Schrodinger Equation ◽  
Wind Forcing ◽  
Nonlinear Schrödinger

The influence of a strong and gusty wind field on ocean waves is investigated. How the random wind affects solitary waves is analyzed in order to obtain insights about wave generation by randomly time varying wind forcing. Using the Euler equations of fluid dynamics and the method of multiple scales, a random nonlinear Schrödinger equation and a random modified nonlinear Schrödinger equation are obtained for randomly wind forced nonlinear deep water waves. Miles theory is used for modeling the pressure variation at the wave surface resulting from the wind velocity field. The nonlinear Schrödinger equation and the modified nonlinear Schrödinger equation are computed using a relaxation pseudo spectral scheme. The results show that the influence of gusty wind on solitary waves leads to a randomly increasing ocean wave envelope. However, in a laboratory setup with much smaller wave amplitudes and higher wave frequencies, the influence of water viscosity is much higher. This leads to fluctuating solutions, which are sensitive to wind forcing.

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 Soliton Turbulence in Approximate and Exact Models for Deep Water Waves

Fluids ◽  
2020 ◽  
Vol 5 (2) ◽  
pp. 67 ◽  
Author(s):  
Dmitry Kachulin ◽  
Alexander Dyachenko ◽  
Vladimir Zakharov
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Water Waves ◽  
Schrodinger Equation ◽  
Modulational Instability ◽  
Nonlinear Schrodinger Equation ◽  
Wave System ◽  
Zakharov Equation ◽  
Nonlinear Schrödinger ◽  
Homogeneous Wave

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.

Sign in / Sign up

Export Citation Format

Share Document