scholarly journals Modulation Instability of Hydro-Elastic Waves Blown by a Wind with a Uniform Vertical Profile

Fluids ◽  
2021 ◽  
Vol 6 (12) ◽  
pp. 458
Author(s):  
Susam Boral ◽  
Trilochan Sahoo ◽  
Yury Stepanyants
Keyword(s):  
Elastic Waves ◽  
Vertical Profile ◽  
Modulation Instability ◽  
Physical Phenomenon ◽  
Membrane Surface ◽  
Multiple Scale ◽  
Negative Energy ◽  
Velocity Versus ◽  
Nls Equation ◽  
Rubber Membrane

An interesting physical phenomenon was recently observed when a fresh-water basin is covered by a thin ice film that has properties similar to the property of a rubber membrane. Surface waves can be generated under the action of wind on the air–water interface that contains an ice film. The modulation property of hydro-elastic waves (HEWs) in deep water covered by thin ice film blown by the wind with a uniform vertical profile is studied here in terms of the airflow velocity versus wavenumber. The modulation instability of HEWs is studied through the analysis of coefficients of the nonlinear Schrödinger (NLS) equation with the help of the Lighthill criterion. The NLS equation is derived using the multiple scale method in the presence of airflow. It is demonstrated that the potentially unstable hydro-elastic waves with negative energy appear for relatively small wind speeds, whereas the Kelvin–Helmholtz instability arises when the wind speed becomes fairly strong. Estimates of parameters of modulated waves for the typical conditions are given.

scholarly journals Modulation Instability of Hydro-Elastic Waves Blown by a Wind with a Uniform Vertical Profile

2021 ◽  
Author(s):  
S. Boral ◽  
T. Sahoo ◽  
Y. Stepanyants
Keyword(s):  
Elastic Waves ◽  
Vertical Profile ◽  
Modulation Instability ◽  
Physical Phenomenon ◽  
Membrane Surface ◽  
Multiple Scale ◽  
Negative Energy ◽  
Velocity Versus ◽  
Nls Equation ◽  
Rubber Membrane

An interesting physical phenomenon was recently observed when a fresh-water basin is covered by a thin ice film that has properties similar to that of a rubber membrane. Surface waves can be generated under the action of wind on the air-water interface that contains an ice film. The modulation property of hydro-elastic waves (HEWs) in deep water covered by thin ice film blown by the wind with a uniform vertical profile is studied here in terms of the air-flow velocity versus a wavenumber. The modulation instability of HEWs is studied through the analysis of coefficients of the nonlinear Schrödinger (NLS) equation with the help of the Lighthill criterion. The NLS equation is derived using the multiple scale method in the presence of airflow. It is demonstrated that the potentially unstable hydro-elastic waves with negative energy appear for relatively small wind speeds, whereas the Kelvin–Helmholtz instability arises when the wind speed becomes fairly strong. Estimates of parameters of modulated waves for the typical conditions are given.

scholarly journals (2 + 1)-Dimensional Coupled Model for Envelope Rossby Solitary Waves and Its Solutions as well as Chirp Effect

2017 ◽  
Vol 2017 ◽  
pp. 1-12
Author(s):  
Xin Chen ◽  
Hongwei Yang ◽  
Min Guo ◽  
Baoshu Yin
Keyword(s):  
Solitary Waves ◽  
Rossby Waves ◽  
Modulation Instability ◽  
Multiple Scales ◽  
Coupled Model ◽  
Nls Equation ◽  
Coupled Models ◽  
Trial Function Method ◽  
Rossby Solitary Waves ◽  
Stable Feature

Using the method of multiple scales and perturbation method, a set of coupled models describing the envelope Rossby solitary waves in (2+1)-dimensional condition are obtained, also can be called coupled NLS (CNLS) equations. Following this, based on trial function method, the solutions of the NLS equation are deduced. Moreover, the modulation instability of coupled envelope Rossby waves is studied. We can find that the stable feature of coupled envelope Rossby waves is decided by the value of S. Finally, learning from the concept of chirp in the optical soliton communication field, we study the chirp effect caused by nonlinearity and dispersion in the propagation of Rossby waves.

scholarly journals Multi-rogue Wave Solutions for a Generalized Integrable Discrete Nonlinear Schrodinger Equation With Higher-order Excitations

2021 ◽  
Author(s):  
Ma Li-Yuan ◽  
Yang Jun ◽  
Zhang Yan-Li
Keyword(s):  
Modulation Instability ◽  
Rogue Wave ◽  
Order Term ◽  
Higher Order ◽  
Discrete Version ◽  
Nls Equation ◽  
Third Order ◽  
Discrete Nonlinear Schrödinger Equation ◽  
Dynamical Behaviors ◽  
Continuous Waves

Abstract In this paper, we construct the discrete rogue wave(RW) solutions for a higher-order or generalized integrable discrete nonlinear Schr¨odinger(NLS) equation. First, based on the modified Lax pair, the discrete version of generalized Darboux transformation are constructed. Second, the dynamical behaviors of first-, second- and third-order RWsolutions are investigated in corresponding to the unique spectral parameter λ, higher-order term coefficient γ, and free constants dk, fk (k = 1, 2, · · · ,N), which exhibit affluent wave structures. The differences between the RW solution of the higher-order discrete NLS equation and that of the Ablowitz-Ladik(AL) equation are illustrated in figures. Moreover, numerical experiments are explored, which demonstrates that strong-interaction RWs are stabler than the weak-interaction RWs. Finally, the modulation instability of continuous waves is studied.

scholarly journals Superregular breathers, characteristics of nonlinear stage of modulation instability induced by higher-order effects

2017 ◽  
Vol 473 (2199) ◽  
pp. 20160681 ◽  
Author(s):  
Jian-Hui Zhang ◽  
Lei Wang ◽  
Chong Liu
Keyword(s):  
Darboux Transformation ◽  
Optical Pulse ◽  
Modulation Instability ◽  
Higher Order ◽  
Order Effects ◽  
Optical Fibres ◽  
Ultrashort Optical Pulse ◽  
Nls Equation ◽  
Nonlinear Stage ◽  
Higher Order Effects

We study the higher-order generalized nonlinear Schrödinger (NLS) equation describing the propagation of ultrashort optical pulse in optical fibres. By using Darboux transformation, we derive the superregular breather solution that develops from a small localized perturbation. This type of solution can be used to characterize the nonlinear stage of the modulation instability (MI) of the condensate. In particular, we show some novel characteristics of the nonlinear stage of MI arising from higher-order effects: (i) coexistence of a quasi-Akhmediev breather and a multipeak soliton; (ii) two multipeak solitons propagation in opposite directions; (iii) a beating pattern followed by two multipeak solitons in the same direction. It is found that these patterns generated from a small localized perturbation do not have the analogues in the standard NLS equation. Our results enrich Zakharov’s theory of superregular breathers and could provide helpful insight on the nonlinear stage of MI in presence of the higher-order effects.

Modulational instability of ion-acoustic waves in fully relativistic two-component plasma

2015 ◽  
Vol 81 (3) ◽  
Author(s):  
B. Ghosh ◽  
S. Banerjee
Keyword(s):  
Growth Rate ◽  
Acoustic Waves ◽  
Relativistic Effect ◽  
Modulational Instability ◽  
Perturbation Technique ◽  
Multiple Scale ◽  
Nls Equation ◽  
Ion Acoustic Waves ◽  
Ion Acoustic ◽  
Component Plasma

Nonlinear amplitude modulation of ion-acoustic waves (IAWs) in a fully relativistic unmagnetized two-fluid plasma has been theoretically studied by using complete set of fully relativistic dynamic equations. To describe the nonlinear evolution of the wave envelope a nonlinear Schrödinger (NLS) equation is derived by using standard multiple scale perturbation technique. Using this equation it is shown that the wave becomes modulationally unstable as the wavenumber exceeds certain critical value. This critical wavenumber is found to decrease with increase in relativistic effect. The instability growth rate has also been calculated numerically for different values of plasma drift velocity. The growth rate is shown to decrease with increase in the relativistic effect.

Resonant interactions between Kelvin ship waves and ambient waves

2008 ◽  
Vol 597 ◽  
pp. 171-197 ◽  
Author(s):  
QIANG ZHU ◽  
YUMING LIU ◽  
DICK K. P. YUE
Keyword(s):  
Wave Vector ◽  
Continuous Wave ◽  
Resonance Condition ◽  
Multiple Scale ◽  
Variable Coefficients ◽  
Kelvin Wave ◽  
Nls Equation ◽  
Near Wake ◽  
Resonance Mechanism ◽  
Ship Waves

We consider the nonlinear interactions between the steady Kelvin waves behind an advancing ship and an (unsteady) ambient wave. It is shown that, for moderately steep ship waves and/or ambient waves, third-order (quartet) resonant interaction among the two wave systems could occur, leading to the generation of a new propagating wave along a specific ray in the Kelvin wake. The wave vector of the generated wave as well as the angle of the resonance ray are determined by the resonance condition and are functions of the ship forward speed and the wave vector of the ambient wave. To understand the resonance mechanism and the characteristics of the generated wave, we perform theoretical analyses of this problem using two related approaches. To obtain a relatively simple model in the form of a nonlinear Schrödinger (NLS) equation for the evolution of the resonant wave, we first consider a multiple-scale approach assuming locally discrete Kelvin wave components, with constant wave vectors but varying amplitudes along the resonance ray. This NLS model captures the key resonance mechanism but does not account for the detuning effect associated with the wave vector variation of Kevin waves in the neighbourhood of the resonance ray. To obtain the full quantitative features and evolution characteristics, we also consider a more complete model based on Zakharov's integral equation applied in the context of a continuous wave vector spectrum. The resulting evolution equation can be reduced to an NLS form with, however, cross-ray variable coefficients, on imposing a narrow-band assumption valid in the neighbourhood of the resonance ray. As expected, the two models compare well when wave vector detuning is small, in the near wake close to the ray. To verify the analyses, direct high-resolution simulations of the nonlinear wave interaction problem are obtained using a high-order spectral method. The simulations capture the salient features of the resonance in the near wake of the ship, with good agreements with theory for the location of the resonance and the growth rate of the generated wave.

Quantum solitons in the Fermi–Pasta–Ulam model

2014 ◽  
Vol 28 (11) ◽  
pp. 1450075 ◽  
Author(s):  
De-Jun Li ◽  
Bing Tang
Keyword(s):  
Fourier Transform ◽  
Inverse Fourier Transform ◽  
Multiple Scale ◽  
Coordinate Space ◽  
Localized Modes ◽  
Nls Equation ◽  
Transform Method ◽  
Intrinsic Localized Modes ◽  
Scale Method ◽  
Quantum Solitons

In this paper, quantum solitons in the Fermi–Pasta–Ulam (FPU) model are investigated analytically. By using the canonical transform method and number-conserving approximation, we obtain the normal form of the phonon-conserving quantized Hamiltonian. In order to convert the quantized Hamiltonian into the coordinate space, we employ the inverse Fourier transform. With the help of the Hartree approximate and the semidiscrete multiple-scale method, the nonlinear Schrödinger (NLS) equation is derived. The results show that quantum solitons may exist in the FPU model. Moreover, it is found that moving quantum solitons become quantum intrinsic localized modes under certain condition. In addition, we obtain the energy level of quantum solitons, which indicates that the energy of such quantum solitons is quantized.

scholarly journals Modulation Instability Analysis of a Nonautonomous (3+1)-Dimensional Coupled Nonlinear Schrödinger Equation

2021 ◽  
Author(s):  
Arvind Patel ◽  
VINEESH KUMAR
Keyword(s):  
Phase Modulation ◽  
Modulation Instability ◽  
Dispersion Coefficient ◽  
Algebraic Function ◽  
Test Functions ◽  
Nls Equation ◽  
Dispersion Phase ◽  
Nonlinear Schrödinger ◽  
Wave Numbers ◽  
Standard Linear

Abstract We investigate the modulation instability (MI) analysis of a nonautonomous (3+1)-dimensional coupled nonlinear Schrödinger (NLS) equation with time-dependent dispersion and phase modulation coefficients. By employing standard linear stability analysis, we obtain an explicit expression for the MI gain as a function of dispersion, phase modulation, perturbation wave numbers and an initial incidence power. The nonautonomous coupled NLS equation is found to be modulationally unstable for the same sign of dispersion and phase modulation coefficients. This equation is modulationally stable for zero dispersion and or phase modulation coefficients. But non-zero dispersion coefficient, it is modulationally stable/unstable on distinct bandwidth of wave numbers. The trigonometric, exponential, algebraic function of time and constant have been chosen as test functions for dispersion and phase modulation to find the effect on the MI analysis. The effect of focusing and defocusing medium on the MI analysis has also been investigated. The MI bandwidth in the focusing medium is found to be larger than defocusing medium. It is found that the modulation instability of the equation can be managed by proper choice of the dispersion and phase modulation parameters.

scholarly journals Modulation Instability of Ion-Acoustic Waves in Plasma with Nonthermal Electrons

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Basudev Ghosh ◽  
Sreyasi Banerjee
Keyword(s):  
Acoustic Waves ◽  
Modulational Instability ◽  
Perturbation Technique ◽  
Collisionless Plasma ◽  
Multiple Scale ◽  
Ion Temperature ◽  
Nls Equation ◽  
Ion Acoustic Waves ◽  
Nonthermal Electrons ◽  
Ion Acoustic

Modulational instability of ion-acoustic waves has been theoretically investigated in an unmagnetized collisionless plasma with nonthermal electrons, Boltzmann positrons, and warm positive ions. To describe the nonlinear evolution of the wave amplitude a nonlinear Schrödinger (NLS) equation has been derived by using multiple scale perturbation technique. The nonthermal parameter, positron concentration, and ion temperature are shown to play significant role in the modulational instability of ion-acoustic waves and the formation of envelope solitons.

Minimisation of biomass in an extractive membrane bioreactor

1996 ◽  
Vol 34 (5-6) ◽  
pp. 273-280 ◽  
Author(s):  
L. F. Strachan ◽  
L. M. Freitas dos Santos ◽  
D. J. Leak ◽  
A. G. Livingston
Keyword(s):  
Membrane Bioreactor ◽  
Energy Requirement ◽  
Membrane Surface ◽  
Growth Conditions ◽  
Organic Substrate ◽  
Rubber Membrane ◽  
Biofilm Growth ◽  
Volatile Organic ◽  
Maintenance Energy Requirement ◽  
Biological Methods

Many traditional biological methods for the treatment of wastewater cope poorly with toxic, volatile organic compounds. The extractive membrane bioreactor is a novel process for the treatment of industrial wastewaters containing such compounds which combines extraction across a silicone rubber membrane with biodegradation. Previous work has shown that there is a problem in this system with excess biofilm growth on the membrane surface, resulting in reduced flux of organic substrate across the membrane. The work presented here shows that addition of sodium chloride to the biomedium increases the maintenance energy requirement of the degradative microorganisms and results, in a carbon-limited situation, in a reduction in biofilm growth. Flux of organic substrate was shown to remain high under reduced biofilm growth conditions.

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