scholarly journals Water waves, nonlinear Schrödinger equations and their solutions

1983 ◽  
Vol 25 (1) ◽  
pp. 16-43 ◽  
Author(s):  
D. H. Peregrine
Keyword(s):  
Water Waves ◽  
Rational Solution ◽  
Three Dimensional ◽  
Length Scale ◽  
Nls Equation ◽  
Nonlinear Water Waves ◽  
Weakly Nonlinear ◽  
Nonlinear Schrödinger ◽  
Wave Induced ◽  
Modulation Length

AbstractEquations governing modulations of weakly nonlinear water waves are described. The modulations are coupled with wave-induced mean flows except in the case of water deeper than the modulation length scale. Equations suitable for water depths of the order the modulation length scale are deduced from those derived by Davey and Stewartson [5] and Dysthe [6]. A number of ases in which these equations reduce to a one dimensional nonlinear Schrödinger (NLS) equation are enumerated.Several analytical solutions of NLS equations are presented, with discussion of some of their implications for describing the propagation of water waves. Some of the solutions have not been presented in detail, or in convenient form before. One is new, a “rational” solution describing an “amplitude peak” which is isolated in space-time. Ma's [13] soli ton is particularly relevant to the recurrence of uniform wave trains in the experiment of Lake et al.[10].In further discussion it is pointed out that although water waves are unstable to three-dimensional disturbances, an effective description of weakly nonlinear two-dimensional waves would be a useful step towards describing ocean wave propagation.

Phenomena on Rogue Waves and Rational Solution of Korteweg-de Vries Equation

2013 ◽  
Author(s):  
Ni Song ◽  
Wei Zhang ◽  
Qian Wang
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Kdv Equation ◽  
Rational Solution ◽  
Rogue Waves ◽  
Nls Equation ◽  
Engineering Structures ◽  
Weakly Nonlinear ◽  
Nonlinear Mechanism ◽  
De Vries

An appropriate nonlinear mechanism may create the rogue waves. Perhaps the simplest mechanism, which is able to create considerate changes in the wave amplitude, is the nonlinear interaction of shallow-water solitons. The most well-known examples of such structure are Korteweg-de Vries (KdV) solitons. The Korteweg-de Vries (KdV) equation, which describes the shallow water waves, is a basic weakly dispersive and weakly nonlinear model. Basing on the homogeneous balanced method, we achieve the general rational solution of a classical KdV equation. Numerical simulations of the solution allow us to explain rare and unexpected appearance of the rogue waves. We compare the rogue waves with the ones generated by the nonlinear Schrödinger (NLS) equation which can describe deep water wave trains. The numerical results illustrate that the amplitude of the KdV equation is higher than the one of the NLS equation, which may causes more serious damage of engineering structures in the ocean. This nonlinear mechanism will provide a theoretical guidance in the ocean and physics.

scholarly journals Nonlinear water waves in channels of arbitrary shape

1992 ◽  
Vol 242 ◽  
pp. 211-233 ◽  
Author(s):  
Michelle H. Teng ◽  
Theodore Y. Wu
Keyword(s):  
Water Waves ◽  
Arbitrary Shape ◽  
Rectangular Channel ◽  
Three Dimensional ◽  
Order Effect ◽  
Equation Model ◽  
Nonlinear Water Waves ◽  
Cross Sectional ◽  
Shape Factors ◽  
Weakly Nonlinear

The generalized channel Boussinesq (gcB) two-equation model and the forced channel Korteweg–de Vries (cKdV) one-equation model previously derived by the authors are further analysed and discussed in the present study. The gcB model describes the propagation and generation of weakly nonlinear, weakly dispersive and weakly forced long water waves in channels of arbitrary shape that may vary both in space and time, and the cKdV model is applicable to unidirectional motions of such waves, which may be sustained under forcing at resonance of the system. These two models are long-wave approximations of a hierarchy set of section-mean conservation equations of mass, momentum and energy, which are exact for inviscid fluids. Results of these models are demonstrated with four specific channel shapes, namely variable rectangular, triangular, parabolic and semicircular sections, in which case solutions are obtained in closed form. In particular, for uniform channels of equal mean water depth, different cross-sectional shapes have a leading-order effect only on the variations of a k-factor of the coefficient of the term bearing the dispersive effects in the model equations. For this case, the uniform-channel analogy theorem enunciated here shows that long waves of equal (mean) height in different uniform channels of equal mean depth but distinct k-shape factors will propagate with equal velocity and with their effective wavelengths appearing k times of that in the rectangular channel, for which k = 1. It also shows that the further channel shape departs from the rectangular, the greater the value of k. Based on this observation, the solitary and cnoidal waves in a k-shaped channel are compared with experiments on wave profiles and wave velocities. Finally, some three-dimensional features of these solitary waves are presented for a triangular channel.

Three-Dimensional Weakly Nonlinear Shallow Water Waves Regime and its Traveling Wave Solutions

2018 ◽  
Vol 15 (03) ◽  
pp. 1850017 ◽  
Author(s):  
Aly R. Seadawy
Keyword(s):  
Free Surface ◽  
Solitary Wave ◽  
Shallow Water ◽  
Traveling Wave ◽  
Water Waves ◽  
Three Dimensional ◽  
Traveling Wave Solutions ◽  
Shallow Water Waves ◽  
Weakly Nonlinear ◽  
Wave Solutions

The problem formulations of models for three-dimensional weakly nonlinear shallow water waves regime in a stratified shear flow with a free surface are studied. Traveling wave solutions are generated by deriving the nonlinear higher order of nonlinear evaluation equations for the free surface displacement. We obtain the velocity potential and pressure fluid in the form of traveling wave solutions of the obtained nonlinear evaluation equation. The obtained solutions and the movement role of the waves of the exact solutions are new travelling wave solutions in different and explicit form such as solutions (bright and dark), solitary wave, periodic solitary wave elliptic function solutions of higher-order nonlinear evaluation equation.

Nonlinear Water Waves Over a Three-Dimensional Porous Bottom Using Boussinesq-Type Model

2005 ◽  
Vol 47 (4) ◽  
pp. 231-253 ◽  
Author(s):  
Shih-Chun Hsiao ◽  
Philip L.-F. Liu ◽  
Hwung-Hweng Hwung ◽  
Seung-Buhm Woo
Keyword(s):  
Water Waves ◽  
Three Dimensional ◽  
Type Model ◽  
Nonlinear Water Waves

Evolution of weakly nonlinear water waves in the presence of viscosity and surfactant

1991 ◽  
Vol 229 (-1) ◽  
pp. 135 ◽  
Author(s):  
S. W. Joo ◽  
A. F. Messiter ◽  
W. W. Schultz
Keyword(s):  
Water Waves ◽  
Nonlinear Water Waves ◽  
Weakly Nonlinear

scholarly journals Model Equations for Three-Dimensional Nonlinear Water Waves under Tangential Electric Field

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
Bo Tao
Keyword(s):  
Electric Field ◽  
Water Waves ◽  
Three Dimensional ◽  
Liquid Sheet ◽  
Capillary Waves ◽  
Uniform Electric Field ◽  
Nonlinear Water Waves ◽  
Petviashvili Equation ◽  
Model Equations ◽  
Layer Problem

We are concerned with gravity-capillary waves propagating on the surface of a three-dimensional electrified liquid sheet under a uniform electric field parallel to the undisturbed free surface. For simplicity, we make an assumption that the permittivity of the fluid is much larger than that of the upper-layer gas; hence, this two-layer problem is reduced to be a one-layer problem. In this paper, we propose model equations in the shallow-water regime based on the analysis of the Dirichlet-Neumann operator. The modified Benney-Luke equation and Kadomtsev-Petviashvili equation will be derived, and the truly three-dimensional fully localized traveling waves, which are known as “lumps” in the literature, are numerically computed in the Benney-Luke equation.

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