Fifth order evolution equation of gravity-capillary waves

2017 ◽  
Vol 59 ◽  
pp. 103
Author(s):  
Dipankar Chowdhury ◽  
Suma Debsarma
Keyword(s):  
Evolution Equation ◽  
Capillary Waves ◽  
Fifth Order

FIFTH-ORDER EVOLUTION EQUATION OF GRAVITY–CAPILLARY WAVES

2017 ◽  
Vol 59 (1) ◽  
pp. 103-114
Author(s):  
DIPANKAR CHOWDHURY ◽  
SUMA DEBSARMA
Keyword(s):  
Evolution Equation ◽  
Evolution Equations ◽  
Capillary Wave ◽  
Wave Train ◽  
Critical Value ◽  
Capillary Waves ◽  
Wave Number ◽  
Region Of Stability ◽  
Fifth Order ◽  
Nonlinear Gravity

We extend the evolution equation for weak nonlinear gravity–capillary waves by including fifth-order nonlinear terms. Stability properties of a uniform Stokes gravity–capillary wave train is studied using the evolution equation obtained here. The region of stability in the perturbed wave-number plane determined by the fifth-order evolution equation is compared with that determined by third- and fourth-order evolution equations. We find that if the wave number of longitudinal perturbations exceeds a certain critical value, a uniform gravity–capillary wave train becomes unstable. This critical value increases as the wave steepness increases.

Odd-viscosity-induced stabilization of viscous thin liquid films

2019 ◽  
Vol 878 ◽  
pp. 169-189
Author(s):  
E. Kirkinis ◽  
A. V. Andreev
Keyword(s):  
Evolution Equation ◽  
Liquid Films ◽  
Solid Substrate ◽  
Capillary Waves ◽  
Thin Liquid Films ◽  
Shear Stresses ◽  
Thermocapillary Effect ◽  
Surface Shear ◽  
Thermocapillary Instability ◽  
Parameter Ranges

Thin viscous liquid films sitting on a solid substrate support nonlinear capillary waves, driven by surface shear stresses at a liquid–gas interface. When surface tension is spatially dependent other mechanisms, such as the thermocapillary effect, influence the dynamics of thin films. In this article we show that in liquids with broken time-reversal symmetry the character of the aforementioned waves and of the thermocapillary effect are significantly modified due to the presence of odd or Hall viscosity in the liquid. This is because odd viscosity gives rise to new terms in the pressure gradient of the flow thus modifying the evolution equation of the liquid–gas interface accordingly. These terms in turn break the reflection symmetry of the evolution equation leading the system to evolve from a pitchfork to a Hopf bifurcation. The odd-viscosity incipient waves can stabilize unstable thin liquid films. For instance, we show that they can suppress the thermocapillary instability. We establish the parameter ranges that odd viscosity has to satisfy in order to initiate those waves that will lead to stability.

Fifth order evolution equation for long wave dissipative solitons

2012 ◽  
Vol 376 (4) ◽  
pp. 452-456 ◽  
Author(s):  
M.C. Depassier ◽  
J.A. Letelier
Keyword(s):  
Evolution Equation ◽  
Long Wave ◽  
Dissipative Solitons ◽  
Fifth Order

scholarly journals On solitons: Propagation of shallow water waves for the fifth-order KdV hierarchy integrable equation

Open Physics ◽  
2021 ◽  
Vol 19 (1) ◽  
pp. 828-842
Author(s):  
Aly R. Seadawy ◽  
Shafiq U. Rehman ◽  
Muhammad Younis ◽  
Syed T. R. Rizvi ◽  
Ali Althobaiti
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Sound Propagation ◽  
Soliton Solutions ◽  
Integrable Equation ◽  
Capillary Waves ◽  
Nonlinear Phenomena ◽  
Rational Solutions ◽  
Shallow Water Waves ◽  
Fifth Order

Abstract This article studies the fifth-order KdV (5KdV) hierarchy integrable equation, which arises naturally in the modeling of numerous wave phenomena such as the propagation of shallow water waves over a flat surface, gravity–capillary waves, and magneto-sound propagation in plasma. Two innovative integration norms, namely, the G ′ G 2 \left(\frac{{G}^{^{\prime} }}{{G}^{2}}\right) -expansion and ansatz approaches, are used to secure the exact soliton solutions of the 5KdV type equations in the shapes of hyperbolic, singular, singular periodic, shock, shock-singular, solitary wave, and rational solutions. The constraint conditions of the achieved solutions are also presented. Besides, by selecting appropriate criteria, the actual portrayal of certain obtained results is sorted out graphically in three-dimensional, two-dimensional, and contour graphs. The results suggest that the procedures used are concise, direct, and efficient, and that they can be applied to more complex nonlinear phenomena.

On asymmetric gravity–capillary solitary waves

1997 ◽  
Vol 330 ◽  
pp. 215-232 ◽  
Author(s):  
T.-S. YANG ◽  
T. R. AKYLAS
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Numerical Solutions ◽  
Solitary Wave Solution ◽  
Kdv Equation ◽  
Phase Speed ◽  
Capillary Waves ◽  
Asymptotic Results ◽  
Water Wave Problem ◽  
Fifth Order

Symme tric gravity–capillary solitary waves with decaying oscillatory tails are known to bifurcate from infinitesimal periodic waves at the minimum value of the phase speed where the group velocity is equal to the phase speed. In the small-amplitude limit, these solitary waves may be interpreted as envelope solitons with stationary crests and are described by the nonlinear Schrödinger (NLS) equation to leading order. In line with this interpretation, it would appear that one may also co nstruct asymmetric solitary waves by shifting the carrier oscillations relative to the envelope of a symmetric solitary wave. This possibility is examined here on the basis of the fifth-order Korteweg–de Vries (KdV) equation, a model for g ravity–capillary waves on water of finite depth when the Bond number is close to 1/3. Using techniques of exponential asymptotics beyond all orders of the NLS theory, it is shown that asymmetric solitary waves of the form suggested by the NLS theory in fact are not possible. On the other hand, an infinity of symmetric and asymmetric solitary-wave solution families comprising two or more NLS solitary wavepackets bifurcate at finite values of the amplitude parameter. The asymptotic results are consistent with numerical solutions of the fifth-order KdV equation. Moreover, the asymptotic theory suggests that such multi-packet gravity–capillary solitary waves also exist in the full water-wave problem near the minimum of t he phase speed.

Similarities in a fifth-order evolution equation with and with no singular kernel

2020 ◽  
Vol 130 ◽  
pp. 109467 ◽  
Author(s):  
Emile F. Doungmo Goufo ◽  
Sunil Kumar ◽  
S.B. Mugisha
Keyword(s):  
Evolution Equation ◽  
Singular Kernel ◽  
Fifth Order
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