scholarly journals Transversally periodic solitary gravity–capillary waves

2014 ◽  
Vol 470 (2161) ◽  
pp. 20130537 ◽  
Author(s):  
Paul A. Milewski ◽  
Zhan Wang
Keyword(s):  
Solitary Waves ◽  
Water Waves ◽  
Travelling Waves ◽  
Plane Waves ◽  
Three Dimensional ◽  
Capillary Waves ◽  
Propagation Direction ◽  
The Stability ◽  
Dirichlet To Neumann Operator ◽  
The Waves

When both gravity and surface tension effects are present, surface solitary water waves are known to exist in both two- and three-dimensional infinitely deep fluids. We describe here solutions bridging these two cases: travelling waves which are localized in the propagation direction and periodic in the transverse direction. These transversally periodic gravity–capillary solitary waves are found to be of either elevation or depression type, tend to plane waves below a critical transverse period and tend to solitary lumps as the transverse period tends to infinity. The waves are found numerically in a Hamiltonian system for water waves simplified by a cubic truncation of the Dirichlet-to-Neumann operator. This approximation has been proved to be very accurate for both two- and three-dimensional computations of fully localized gravity–capillary solitary waves. The stability properties of these waves are then investigated via the time evolution of perturbed wave profiles.

scholarly journals Dynamics of gravity–capillary solitary waves in deep water

2012 ◽  
Vol 708 ◽  
pp. 480-501 ◽  
Author(s):  
Zhan Wang ◽  
Paul A. Milewski
Keyword(s):  
Solitary Waves ◽  
Water Waves ◽  
Periodic Structures ◽  
Time Integration ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Full Potential ◽  
Numerical Time Integration ◽  
The Stability ◽  
Nonlinear Focusing

AbstractThe dynamics of solitary gravity–capillary water waves propagating on the surface of a three-dimensional fluid domain is studied numerically. In order to accurately compute complex time-dependent solutions, we simplify the full potential flow problem by using surface variables and taking a particular cubic truncation possessing a Hamiltonian with desirable properties. This approximation agrees remarkably well with the full equations for the bifurcation curves, wave profiles and the dynamics of solitary waves for a two-dimensional fluid domain, and with higher-order truncations in three dimensions. Fully localized solitary waves are then computed in the three-dimensional problem and the stability and interaction of both line and localized solitary waves are investigated via numerical time integration of the equations. There are many solitary wave branches, indexed by their finite energy as their amplitude tends to zero. The dynamics of the solitary waves is complex, involving nonlinear focusing of wavepackets, quasi-elastic collisions, and the generation of propagating, spatially localized, time-periodic structures akin to breathers.

Stability and instability of some nonlinear dispersive solitary waves in higher dimension

1996 ◽  
Vol 126 (1) ◽  
pp. 89-112 ◽  
Author(s):  
Anne de Bouard
Keyword(s):  
Solitary Waves ◽  
Acoustic Waves ◽  
Vries Equation ◽  
Three Dimensional ◽  
Higher Dimension ◽  
Ion Acoustic Waves ◽  
Stability And Instability ◽  
Ion Acoustic ◽  
De Vries ◽  
The Stability

We study the stability of positive radially symmetric solitary waves for a three dimensional generalisation of the Korteweg de Vries equation, which describes nonlinear ion-acoustic waves in a magnetised plasma, and for a generalisation in dimension two of the Benjamin–Bona–Mahony equation.

On the subharmonic instabilities of steady three-dimensional deep water waves

1994 ◽  
Vol 262 ◽  
pp. 265-291 ◽  
Author(s):  
Mansour Ioualalen ◽  
Christian Kharif
Keyword(s):  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Transverse Direction ◽  
Plane Waves ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Two Dimensional ◽  
Wave Steepness ◽  
Oblique Angle

A numerical procedure has been developed to study the linear stability of nonlinear three-dimensional progressive gravity waves on deep water. The three-dimensional patterns considered herein are short-crested waves which may be produced by two progressive plane waves propagating at an oblique angle, γ, to each other. It is shown that for moderate wave steepness the dominant resonances are sideband-type instabilities in the direction of propagation and, depending on the value of γ, also in the transverse direction. It is also shown that three-dimensional progressive gravity waves are less unstable than two-dimensional progressive gravity waves.

Experiments on strong interactions between solitary waves

1978 ◽  
Vol 85 (3) ◽  
pp. 417-431 ◽  
Author(s):  
P. D. Weidman ◽  
T. Maxworthy
Keyword(s):  
Shallow Water ◽  
Solitary Waves ◽  
Water Waves ◽  
Vries Equation ◽  
Rectangular Channel ◽  
Quantitative Agreement ◽  
Strong Interactions ◽  
Shallow Water Waves ◽  
Qualitative And Quantitative ◽  
The Waves

Experiments on the interaction between solitary shallow-water waves propagating in the same direction have been performed in a rectangular channel. Two methods were devised to compensate for the dissipation of the waves in order to compare results with Hirota's (1971) solution for the collision of solitons described by the Kortewegde Vries equation. Both qualitative and quantitative agreement with theory is obtained using the proposed corrections for wave damping.

On analyticity of travelling water waves

2005 ◽  
Vol 461 (2057) ◽  
pp. 1283-1309 ◽  
Author(s):  
David P Nicholls ◽  
Fernando Reitich
Keyword(s):  
Water Waves ◽  
Bifurcation Theory ◽  
Amplitude Ratio ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Perturbation Parameter ◽  
Boundary Perturbation ◽  
Classical Boundary ◽  
Boundary Model ◽  
Special Case

In this paper we establish the existence and analyticity of periodic solutions of a classical free-boundary model of the evolution of three-dimensional, capillary–gravity waves on the surface of an ideal fluid. The result is achieved through the application of bifurcation theory to a boundary perturbation formulation of the problem, and it yields analyticity jointly with respect to the perturbation parameter and the spatial variables. The travelling waves we find can be interpreted as resulting from the (nonlinear) interaction of two two-dimensional wavetrains, giving rise to a periodic travelling pattern. Our analyticity theorem extends the most sophisticated results known to date in the absence of resonance; ‘short crested waves’, which result from the interaction of two wavetrains with unit amplitude ratio are realized as a special case. Our method of proof also sheds light on the convergence and conditioning properties of classical boundary perturbation methods for the numerical approximation of travelling surface waves. Indeed, we demonstrate that the rather unstable numerical behaviour of these approaches can be attributed to the strong but subtle cancellations in the formulas underlying their classical implementations. These observations motivate the derivation and use of an alternative, stable, formulation which, in addition to providing our method of proof, suggests new stabilized implementations of boundary perturbation algorithms.

scholarly journals Existence and conditional energetic stability of solitary gravity–capillary water waves with constant vorticity

2015 ◽  
Vol 145 (4) ◽  
pp. 791-883 ◽  
Author(s):  
M. D. Groves ◽  
E. Wahlén
Keyword(s):  
Surface Tension ◽  
Water Waves ◽  
Applied Mathematics ◽  
Finite Depth ◽  
Constant Vorticity ◽  
Existence And Stability ◽  
Strong Surface ◽  
Weak Surface ◽  
The Stability ◽  
The Waves

We present an existence and stability theory for gravity–capillary solitary waves with constant vorticity on the surface of a body of water of finite depth. Exploiting a rotational version of the classical variational principle, we prove the existence of a minimizer of the wave energy𝓗subject to the constraint𝓘= 2µ, where𝓘is the wave momentum and 0 <µ≪ 1. Since𝓗and𝓘are both conserved quantities, a standard argument asserts the stability of the setDµof minimizers: solutions starting nearDµremain close toDµin a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water waves of the present kind are described by solutions of a Korteweg–de Vries equation (for strong surface tension) or a nonlinear Schrödinger equation (for weak surface tension). We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of the appropriate model equation asµ↓ 0.

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.

scholarly journals Numerical approximation to Benjamin type equations. Generation and stability of solitary waves

2020 ◽  
Author(s):  
VA Dougalis ◽  
A Duran ◽  
Dimitrios Mitsotakis
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Numerical Approximation ◽  
Solitary Wave Solutions ◽  
Time Stepping ◽  
Runge Kutta Method ◽  
Stability Of Solitary Waves ◽  
The Stability ◽  
Numerical Generation ◽  
The Waves

© 2018 Elsevier B.V. This paper is concerned with the study, by computational means, of the generation and stability of solitary-wave solutions of generalized versions of the Benjamin equation. The numerical generation of the solitary-wave profiles is accurately performed with a modified Petviashvili method which includes extrapolation to accelerate the convergence. In order to study the dynamics of the solitary waves the equations are discretized in space with a Fourier pseudospectral collocation method and a fourth-order, diagonally implicit Runge–Kutta method of composition type as time-stepping integrator. The stability of the waves is numerically studied by performing experiments with small and large perturbations of the solitary pulses as well as interactions of solitary waves.

Experiments on nonlinear gravity–capillary waves

1999 ◽  
Vol 380 ◽  
pp. 205-232 ◽  
Author(s):  
LEV SHEMER ◽  
MELAD CHAMESSE
Keyword(s):  
Narrow Band ◽  
Numerical Study ◽  
Three Dimensional ◽  
Capillary Waves ◽  
Carrier Wave ◽  
Nls Equation ◽  
Zakharov Equation ◽  
The Stability ◽  
Nonlinear Gravity

Benjamin–Feir instability of nonlinear gravity–capillary waves is studied experimentally. The experimental results are compared with computations performed for values of wavelength and steepness identical to those employed in the experiments. The theoretical approach is based on the Zakharov nonlinear equation which is modified here to incorporate weak viscous dissipation. Experiments are performed in a wave ume which has an accurately controlled wavemaker for generation of the carrier wave, as well as an additional independent conical wavemaker for generation of controlled three-dimensional disturbances. The approach adopted in the present experimental investigation allows therefore the determination of the actual boundaries of the instability domain, and not just the most unstable disturbances. Instantaneous surface elevation measurements are performed with capacitance-type wave gauges. Multipoint measurements make it possible to determine the angular dependence of the amplitude of the forced and unforced disturbances, as well as their variation along the tank. The limits of the instability domains obtained experimentally for each set of carrier wave parameters agree favourably with those computed numerically using the model equation. The numerical study shows that application of the Zakharov equation, which is free of the narrow-band approximation adopted in the derivation of the nonlinear Schrödinger (NLS) equation, may lead to qualitatively different results regarding the stability of nonlinear gravity–capillary waves. The present experiments support the results of the numerical investigation.

Bifurcating bright and dark solitary waves for the perturbed cubic-quintic nonlinear Schrödinger equation

1998 ◽  
Vol 128 (3) ◽  
pp. 585-629 ◽  
Author(s):  
Todd Kapitula
Keyword(s):  
Schrödinger Equation ◽  
Solitary Waves ◽  
Schrodinger Equation ◽  
Three Dimensional ◽  
Small Error ◽  
Heteroclinic Cycle ◽  
Perturbation Techniques ◽  
Ginzburg Landau ◽  
The Waves ◽  
Quintic Nonlinear Schrödinger Equation

The existence of bright and dark multi-bump solitary waves for Ginzburg–Landau type perturbations of the cubic-quintic Schrodinger equation is considered. The waves in question are not perturbations of known analytic solitary waves, but instead arise as a bifurcation from a heteroclinic cycle in a three-dimensional ODE phase space. Using geometric singular perturbation techniques, regions in parameter space for which 1-bump bright and dark solitary waves will bifurcate are identified. The existence of N-bump dark solitary waves (N ≧ 1) is shown via an application of the Exchange Lemma with Exponentially Small Error. N-bump bright solitary waves are shown to exist as a consequence of the work of Kapitula and Maier-Paape.

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