Parametrically excited solitary waves

1984 ◽  
Vol 148 ◽  
pp. 451-460 ◽  
Author(s):  
John W. Miles
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Complex Amplitude ◽  
Clean Water ◽  
Vertical Oscillation ◽  
Capillary Length ◽  
Cubic Schrödinger Equation ◽  
Limiting Case ◽  
Long Channel ◽  
Weak Damping

A modulated cross-wave of resonant frequencyω1, carrier frequencyω =ω1 {1 + O(ε)}, slowly varying complex amplitude O(ε½b), longitudinal scale b/ε½ and timescale 1/εω is induced in a long channel of breadth b that contains water of depth d and is subjected to a vertical oscillation of amplitude O(εb) and frequency 2ω, where 0 < ε [Lt ] 1. The complex amplitude satisfies a cubic Schrödinger equation, generalized to incorporate weak damping and the parametric excitation. A solution is obtained that describes the standing solitary wave observed by Wu, Keolian & Rudnick (1984). The results depend on both d/b and l*/b, where l* is the capillary length (l* = 2.7 mm for clean water), and solitary waves are impossible if d/b < 0.325 for l*/b = 0 or if l*/b > 0.045 for d/b [gsim ] 1. The corresponding cnoidal waves (of which the solitary wave is a limiting case) are considered in an appendix.

Bifurcations and nonlinear dynamics of surface waves in Faraday resonance

1995 ◽  
Vol 284 ◽  
pp. 341-358 ◽  
Author(s):  
H. Friedel ◽  
E. W. Laedke ◽  
K. H. Spatschek
Keyword(s):  
Nonlinear Dynamics ◽  
Solitary Wave ◽  
Resonant Frequency ◽  
Coherent Structures ◽  
Nonlinear Waves ◽  
Vertical Oscillation ◽  
Spatial Structures ◽  
Spatial Chaos ◽  
Long Channel ◽  
Solitary Form

The nonlinear dynamics of nonlinear modulated cross-waves of resonant frequency ω1 and carrier frequency ω ≈ ω1 is investigated. In a long channel of width b, that contains fluid of depth d and which is subjected to a vertical oscillation of frequency 2ω, the wave can appear in solitary form. As has been shown previously, the solitary wave is only stable in a certain parameter regime; depending on damping and driving amplitudes the wave becomes unstable. The nonlinear development of the instabilities of solitary waves is the central problem of this paper. It is shown how instabilities are saturated following generic routes to chaos in time with spatially coherent structures. Finally, the case of time-modulated driving amplitudes is also considered. In most cases it appears that nonlinear waves of simple spatial structures take part in the nonlinear dynamics, but a few cases of spatial chaos are also reported.

Comment on ‘Propagation of solitary waves and shock wavelength in the pair plasma (J. Plasma Phys. 78, 525–529, 2012)’

2014 ◽  
Vol 80 (3) ◽  
pp. 513-516
Author(s):  
Frank Verheest
Keyword(s):  
Shock Wave ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Plasma Phys ◽  
Theoretical Underpinning ◽  
Pair Plasma ◽  
Electrons And Positrons ◽  
Small Dissipation ◽  
Pseudopotential Approach ◽  
Wave Properties

In a recent paper ‘Propagation of solitary waves and shock wavelength in the pair plasma (J. Plasma Phys. 78, 525–529, 2012)’, Malekolkalami and Mohammadi investigate nonlinear electrostatic solitary waves in a plasma comprising adiabatic electrons and positrons, and a stationary ion background. The paper contains two parts: First, the solitary wave properties are discussed through a pseudopotential approach, and then the influence of a small dissipation is intuitively sketched without theoretical underpinning. Small dissipation is claimed to lead to a shock wave whose wavelength is determined by linear oscillator analysis. Unfortunately, there are errors and inconsistencies in both the parts, and their combination is incoherent.

scholarly journals Existence and linearized stability of solitary waves for a quasilinear Benney system

2016 ◽  
Vol 146 (3) ◽  
pp. 547-564 ◽  
Author(s):  
João-Paulo Dias ◽  
Mário Figueira ◽  
Filipe Oliveira
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Compact Support ◽  
Travelling Waves ◽  
Standing Waves ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Linearized Stability ◽  
Stability Of Solitary Waves ◽  
Image Position

We prove the existence of solitary wave solutions to the quasilinear Benney systemwhere , –1 < p < +∞ and a, γ > 0. We establish, in particular, the existence of travelling waves with speed arbitrarily large if p < 0 and arbitrarily close to 0 if . We also show the existence of standing waves in the case , with compact support if – 1 < p < 0. Finally, we obtain, under certain conditions, the linearized stability of such solutions.

Observation of a standing kink cross wave parametrically excited

1991 ◽  
Vol 434 (1891) ◽  
pp. 435-440 ◽  
Keyword(s):  
Solitary Waves ◽  
Theoretical Explanation ◽  
Vertical Oscillation ◽  
Cylindrical Tank ◽  
Hyperbolic Tangent ◽  
Parametrically Excited ◽  
Short Side ◽  
Kink Wave ◽  
Forced Waves ◽  
Wave Properties

A layer of water in a cylindrical tank is known to be capable of sustaining standing solitary waves within a certain parametric domain when the tank is excited under vertical oscillation. A new mode of forced waves is discovered to exist in a different parametric domain for rectangular tanks with the wave sloshing across the short side of the tank and with its profile modulated by one or more hyperbolic-tangent, or kink-wave-like envelopes. A theoretical explanation for the kink wave properties is provided. Experiments were performed to confirm their existence.

SOLITARY WAVES OF THE NONLINEAR KLEIN-GORDON EQUATION COUPLED WITH THE MAXWELL EQUATIONS

2002 ◽  
Vol 14 (04) ◽  
pp. 409-420 ◽  
Author(s):  
VIERI BENCI ◽  
DONATO FORTUNATO FORTUNATO
Keyword(s):  
Electromagnetic Field ◽  
Solitary Wave ◽  
Electric Field ◽  
Solitary Waves ◽  
Maxwell Equations ◽  
Gordon Equation ◽  
Klein Gordon ◽  
Klein Gordon Equation

This paper is divided in two parts. In the first part we construct a model which describes solitary waves of the nonlinear Klein-Gordon equation interacting with the electromagnetic field. In the second part we study the electrostatic case. We prove the existence of infinitely many pairs (ψ, E), where ψ is a solitary wave for the nonlinear Klein-Gordon equation and E is the electric field related to ψ.

Stability of ion–acoustic solitary waves in a multi-species magnetized plasma consisting of non-thermal and isothermal electrons

2009 ◽  
Vol 75 (5) ◽  
pp. 593-607 ◽  
Author(s):  
SK. ANARUL ISLAM ◽  
A. BANDYOPADHYAY ◽  
K. P. DAS
Keyword(s):  
Stability Analysis ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Perturbation Expansion ◽  
Magnetized Plasma ◽  
Wave Number ◽  
First Order ◽  
Zk Equation ◽  
Ion Acoustic Solitary Waves ◽  
Ion Acoustic

AbstractA theoretical study of the first-order stability analysis of an ion–acoustic solitary wave, propagating obliquely to an external uniform static magnetic field, has been made in a plasma consisting of warm adiabatic ions and a superposition of two distinct populations of electrons, one due to Cairns et al. and the other being the well-known Maxwell–Boltzmann distributed electrons. The weakly nonlinear and the weakly dispersive ion–acoustic wave in this plasma system can be described by the Korteweg–de Vries–Zakharov–Kuznetsov (KdV-ZK) equation and different modified KdV-ZK equations depending on the values of different parameters of the system. The nonlinear term of the KdV-ZK equation and the different modified KdV-ZK equations is of the form [φ(1)]ν(∂φ(1)/∂ζ), where ν = 1, 2, 3, 4; φ(1) is the first-order perturbed quantity of the electrostatic potential φ. For ν = 1, we have the usual KdV-ZK equation. Three-dimensional stability analysis of the solitary wave solutions of the KdV-ZK and different modified KdV-ZK equations has been investigated by the small-k perturbation expansion method of Rowlands and Infeld. For ν = 1, 2, 3, the instability conditions and the growth rate of instabilities have been obtained correct to order k, where k is the wave number of a long-wavelength plane-wave perturbation. It is found that ion–acoustic solitary waves are stable at least at the lowest order of the wave number for ν = 4.

A regularized model for strongly nonlinear internal solitary waves

2009 ◽  
Vol 629 ◽  
pp. 73-85 ◽  
Author(s):  
WOOYOUNG CHOI ◽  
RICARDO BARROS ◽  
TAE-CHANG JO
Keyword(s):  
Stability Analysis ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Wave Solution ◽  
Wave Amplitude ◽  
Solitary Wave Solution ◽  
Shear Instability ◽  
Internal Solitary Waves ◽  
Strongly Nonlinear ◽  
Regularized Model

The strongly nonlinear long-wave model for large amplitude internal waves in a two-layer system is regularized to eliminate shear instability due to the wave-induced velocity jump across the interface. The model is written in terms of the horizontal velocities evaluated at the top and bottom boundaries instead of the depth-averaged velocities, and it is shown through local stability analysis that internal solitary waves are locally stable to perturbations of arbitrary wavelengths if the wave amplitudes are smaller than a critical value. For a wide range of depth and density ratios pertinent to oceanic conditions, the critical wave amplitude is close to the maximum wave amplitude and the regularized model is therefore expected to be applicable to the strongly nonlinear regime. The regularized model is solved numerically using a finite-difference method and its numerical solutions support the results of our linear stability analysis. It is also shown that the solitary wave solution of the regularized model, found numerically using a time-dependent numerical model, is close to the solitary wave solution of the original model, confirming that the two models are asymptotically equivalent.

scholarly journals Enhanced diapycnal mixing by polarity-reversing internal solitary waves in the South China Sea

2021 ◽  
Author(s):  
Yi Gong ◽  
Haibin Song ◽  
Zhongxiang Zhao ◽  
Yongxian Guan ◽  
Kun Zhang ◽  
...  
Keyword(s):  
South China Sea ◽  
Solitary Wave ◽  
Solitary Waves ◽  
South China ◽  
The South China Sea ◽  
Internal Solitary Wave ◽  
Internal Solitary Waves ◽  
Diapycnal Mixing ◽  
Dongsha Atoll ◽  
Diapycnal Diffusivity

Abstract. Shoaling internal solitary waves near the Dongsha Atoll in the South China Sea dissipate their energy and thus enhance diapycnal mixing, which have an important impact on the oceanic environment and primary productivity. The enhanced diapycnal mixing is patchy and instantaneous. Evaluating its spatiotemporal distribution requires comprehensive observation data. Fortunately, seismic oceanography meets the requirements, thanks to its high spatial resolution and large spatial range. In this paper, we studied three internal solitary waves in reversing polarity near the Dongsha Atoll, and calculated the spatial distribution of resultant diapycnal diffusivity. Our results show that the average diffusivities along three survey lines are two orders of magnitude larger than the open-ocean value. The average diffusivity in the internal solitary wave with reversing polarity is three times that of the non-polarity-reversal region. The diapycnal diffusivity is higher at the front of one internal solitary wave, and gradually decreases from shallow to deep water in the vertical direction. Our results also indicates that (1) the enhanced diapycnal diffusivity is related to reflection seismic events; (2) convective instability and shear instability may both contribute to the enhanced diapycnal mixing in the polarity-reversing process; and (3) the difference between our and previous diffusivity profiles is about 2–3 orders of magnitude, but their vertical distribution is almost the same.

Statistical properties of the internal solitary wave ensemble

2021 ◽  
Author(s):  
Tatiana Talipova ◽  
Ekaterina Didenkulova ◽  
Anna Kokorina ◽  
Efim Pelinovsky
Keyword(s):  
Wave Propagation ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Internal Solitary Wave ◽  
Statistical Moments ◽  
Nonlinear Propagation ◽  
Positive Sign ◽  
Cubic Nonlinearity ◽  
Gardner Equation ◽  
Water Stratification

&lt;p&gt;Internal solitary wave ensembles are often observed on the ocean shelves. The long internal baroclinic tide is generated by a barotropic tide on the shelf edges, and then transforms into the soliton-like wave packets during the nonlinear propagation to the beach. The tide is a periodic process and the solitary wave ensemble appears on the shelf usually each semi-diurnal period of 12.4 hours. This process is very sensitive to the variation of the tide characteristics and the hydrology.&lt;/p&gt;&lt;p&gt;We study the propagation of the soliton ensembles numerically in the framework of the spatial form of the Gardner equation (i.e., the Korteweg-de Vries equation with both, quadratic and cubic nonlinearities) assuming horizontally uniform background and applying periodic conditions in time. The water stratification and the local depth are taken similar to the conditions of the north-western Australian shelf, where the stratification admits the existence of solitons but not breathers. The numerical simulation is performed using the Gardner equation with the negative sign of the cubic nonlinearity. For the study of the statistic properties of the solitary waves we use the ensemble of 50 realizations with the same set of 13 solitary waves which are located randomly. The histograms of the wave amplitudes change as the waves travel. The histogram variations become significant after 50 km of the wave propagation. The third (skewness) and the fourth (kurtosis) statistical moments are computed versus the travel distance. It is shown that the both moments decrease by 20% when the solitary wave groups travel for about 150 km.&lt;/p&gt;&lt;p&gt;A similar simulation is conducted for a variable background within the framework of the variable-coefficient Gardner equation. At some location the water stratification corresponds to the positive sign of the local coefficient of the cubic nonlinearity, and then internal breathers may exist. The wave propagation in horizontally inhomogeneous hydrology leads to the occurrence of complicated patterns of solitons and breathers; in the course of the transformation they can disintegrate or form internal rogue waves. Under these conditions the statistical moments of the wave field are essentially different from case when the breather-like waves cannot occur.&lt;/p&gt;&lt;p&gt;The research was supported by the RFBR grants No 19-05-00161 (TT and EP) and 19-35-60022 (ED). The Foundation for the Advancement of Theoretical Physics and Mathematics &amp;#8220;BASIS&amp;#8221; (&amp;#8470; 20-1-3-3-1) is also acknowledged by ED&lt;/p&gt;

scholarly journals The transformation of an interfacial solitary wave of elevation at a bottom step

2009 ◽  
Vol 16 (1) ◽  
pp. 33-42 ◽  
Author(s):  
V. Maderich ◽  
T. Talipova ◽  
R. Grimshaw ◽  
E. Pelinovsky ◽  
B. H. Choi ◽  
...  
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Numerical Study ◽  
Nonlinear Effects ◽  
Ocean Model ◽  
Gardner Equation ◽  
Fully Nonlinear ◽  
Layer Flow ◽  
The One ◽  
Bottom Step

Abstract. In this paper we study the transformation of an internal solitary wave at a bottom step in the framework of two-layer flow, for the case when the interface lies close to the bottom, and so the solitary waves are elevation waves. The outcome is the formation of solitary waves and dispersive wave trains in both the reflected and transmitted fields. We use a two-pronged approach, based on numerical simulations of the fully nonlinear equations using a version of the Princeton Ocean Model on the one hand, and a theoretical and numerical study of the Gardner equation on the other hand. In the numerical experiments, the ratio of the initial wave amplitude to the layer thickness is varied up one-half, and nonlinear effects are then essential. In general, the characteristics of the generated solitary waves obtained in the fully nonlinear simulations are in reasonable agreement with the predictions of our theoretical model, which is based on matching linear shallow-water theory in the vicinity of a step with solutions of the Gardner equation for waves far from the step.

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