scholarly journals Nonlinear Evolution Equations for Broader Bandwidth Wave Packets in Crossing Sea States

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
S. Debsarma ◽  
S. Senapati ◽  
K. P. Das
Keyword(s):  
Growth Rate ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Modulational Instability ◽  
Wave Packets ◽  
Nonlinear Evolution Equations ◽  
Wave Spectra ◽  
Surface Gravity Wave ◽  
Freak Waves ◽  
Wave Trains

Two coupled nonlinear equations are derived describing the evolution of two broader bandwidth surface gravity wave packets propagating in two different directions in deep water. The equations, being derived for broader bandwidth wave packets, are applicable to more realistic ocean wave spectra in crossing sea states. The two coupled evolution equations derived here have been used to investigate the instability of two uniform wave trains propagating in two different directions. We have shown in figures the behaviour of the growth rate of instability of these uniform wave trains for unidirectional as well as for bidirectional perturbations. The figures drawn here confirm the fact that modulational instability in crossing sea states with broader bandwidth wave packets can lead to the formation of freak waves.

scholarly journals Fourth-order nonlinear evolution equations for surface gravity waves in the presence of a thin thermocline

1997 ◽  
Vol 39 (2) ◽  
pp. 214-229 ◽  
Author(s):  
Sudebi Bhattacharyya ◽  
K. P. Das
Keyword(s):  
Growth Rate ◽  
Fourth Order ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Maximum Growth ◽  
Single Equation ◽  
Maximum Growth Rate ◽  
Nonlinear Evolution Equations ◽  
Wave Number ◽  
Coupled Equations

AbstractTwo coupled nonlinear evolution equations correct to fourth order in wave steepness are derived for a three-dimensional wave packet in the presence of a thin thermocline. These two coupled equations are reduced to a single equation on the assumption that the space variation of the amplitudes takes place along a line making an arbitrary fixed angle with the direction of propagation of the wave. This single equation is used to study the stability of a uniform wave train. Expressions for maximum growth rate of instability and wave number at marginal stability are obtained. Some of the results are shown graphically. It is found that a thin thermocline has a stabilizing influence and the maximum growth rate of instability decreases with the increase of thermocline depth.

EFFECT OF UNIFORM WIND FLOW ON MODULATIONAL INSTABILITY OF TWO CROSSING WAVES OVER FINITE DEPTH WATER

2018 ◽  
Vol 60 (1) ◽  
pp. 118-136
Author(s):  
SUMANA KUNDU ◽  
SUMA DEBSARMA ◽  
K. P. DAS
Keyword(s):  
Growth Rate ◽  
Evolution Equations ◽  
Modulational Instability ◽  
Finite Depth ◽  
Wind Flow ◽  
Water Medium ◽  
Third Order ◽  
Wave Trains ◽  
Air Flow Velocity ◽  
Depth Water

The effect of uniform wind flow on modulational instability of two crossing waves is studied here. This is an extension of an earlier work to the case of a finite-depth water body. Evolution equations are obtained as a set of three coupled nonlinear equations correct up to third order in wave steepness. Figures presented in this paper display the variation in the growth rate of instability of a pair of obliquely interacting uniform wave trains with respect to the changes in the air-flow velocity, depth of water medium and the angle between the directions of propagation of the two wave packets. We observe that the growth rate of instability increases with the increase in the wind velocity and the depth of water medium. It also increases with the decrease in the angle of interaction of the two wave systems.

Modulation instability of weakly nonlinear long internal wave packets

2020 ◽  
Author(s):  
Tatiana Talipova ◽  
Efim Pelinovsky
Keyword(s):  
Evolution Equations ◽  
Nonlinear Term ◽  
Modulation Instability ◽  
Modulational Instability ◽  
Wave Packets ◽  
Nonlinear Evolution Equations ◽  
Density Stratification ◽  
Gardner Equation ◽  
Weakly Nonlinear ◽  
Whitham Equations

<p>We exam the problem of the modulation instability of long internal waves. Such weakly nonlinear weakly dispersive wave packets in one-modal approximation are described by the Gardner equation (Korteweg-de Vries equation with both, quadratic and cubic nonlinearity and necessity condition for modulation instability of such quasi-harmonic waves is the positive coefficient of cubic nonlinear term, which is realized for certain density stratification. Nevertheless the linear dispersive relation used within the Gardner equation is valid for very long waves and does not describe waves of moderate length. It is why some other nonlinear evolution equations are applied in the theory of long surface waves like the Benjamin-Bona-Mahony (BBM) and Whitham equations. We use the extended versions of these equations including cubic nonlinear term and express all  coefficients through modal functions and density stratification. Then, the modulational instability of weakly modulated wave packets is investigated after deriving the nonlinear Schrodinger equation. Improved dispersion relation influences on the increment and size of modulational instability. Obtained results are compared with those, which known within the Gardner model.</p>

Traveling wave solutions for the Couple Boiti-Leon-Pempinelli System by using extended Jacobian elliptic function expansion method

2015 ◽  
Vol 11 (3) ◽  
pp. 3134-3138 ◽  
Author(s):  
Mostafa Khater ◽  
Mahmoud A.E. Abdelrahman
Keyword(s):  
Elliptic Function ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Jacobian Elliptic Function ◽  
Function Expansion

In this work, an extended Jacobian elliptic function expansion method is pro-posed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to the Couple Boiti-Leon-Pempinelli System which plays an important role in mathematical physics.

scholarly journals The unified method for abundant soliton solutions of local time fractional nonlinear evolution equations

2021 ◽  
Vol 22 ◽  
pp. 103979
Author(s):  
Nauman Raza ◽  
Muhammad Hamza Rafiq ◽  
Melike Kaplan ◽  
Sunil Kumar ◽  
Yu-Ming Chu
Keyword(s):  
Local Time ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
The Unified Method ◽  
Unified Method
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