On the asymptotic solutions of the KdV equation with higher-order corrections

Nonlinearity ◽  
2005 ◽  
Vol 18 (4) ◽  
pp. 1443-1461 ◽  
Author(s):  
Georgy I Burde
Keyword(s):  
Kdv Equation ◽  
Higher Order ◽  
Asymptotic Solutions ◽  
The Kdv Equation ◽  
Higher Order Corrections

Higher-order corrections to the ion-acoustic waves in a relativistic plasma (isothermal case)

1988 ◽  
Vol 40 (2) ◽  
pp. 359-367 ◽  
Author(s):  
Gobinda Pada Pakira ◽  
A. Roy Chowdhury ◽  
S. N. Paul
Keyword(s):  
Acoustic Waves ◽  
Kdv Equation ◽  
Perturbation Technique ◽  
Type Equation ◽  
Higher Order ◽  
Relativistic Plasma ◽  
Singular Perturbation Technique ◽  
Ion Acoustic ◽  
Physical Profile ◽  
Higher Order Corrections

As a continuation of our earlier work, we have analysed the higher-order perturbative corrections to the formation of (ion-acoustic) solitary waves in a relativistic plasma. It is found that the relativistic considerations affect the amplitude and width variation - as conjectured in our previous paper. Our analysis employs a higher-order singular perturbation technique, with the elimination of secular terms in stages. In this way we arrive at an inhomogeneous KdV-type equation, which is then solved exactly. At this point a new phenomena arises at a critical value of the phase velocity at which the coefficient of the nonlinear term in the KdV equation vanishes. A new set of stretched co-ordinate is then used to derive a modified KdV equation. In both cases we have numerically computed the specific physical profile of the new solitary wave and its width.

Effect of Higher-Order Corrections on the Propagation of Nonlinear Dust-Acoustic Solitary Waves in Mesospheric Dusty Plasmas

2006 ◽  
Vol 61 (7-8) ◽  
pp. 316-322 ◽  
Author(s):  
Sayed A. Elwakil ◽  
Mohamed T. Attia ◽  
Mohsen A. Zahran ◽  
Emad K. El-Shewy ◽  
Hesham G. Abdelwahed
Keyword(s):  
Acoustic Waves ◽  
Kdv Equation ◽  
Type Equation ◽  
Dusty Plasmas ◽  
Higher Order ◽  
Reductive Perturbation Method ◽  
Dust Acoustic Solitary Waves ◽  
Dust Acoustic ◽  
Renormalization Method ◽  
Higher Order Corrections

The contribution of the higher-order correction to nonlinear dust-acoustic waves are studied using the reductive perturbation method in an unmagnetized collisionless mesospheric dusty plasma. A Korteweg - de Vries (KdV) equation that contains the lowest-order nonlinearity and dispersion is derived from the lowest order of perturbation, and a linear inhomogeneous (KdV-type) equation that accounts for the higher-order nonlinearity and dispersion is obtained. A stationary solution is achived via renormalization method

The Effect of Higher-Order Corrections on the Propagation of Nonlinear Dust-Acoustic Solitary Waves in a Dusty Plasma with Nonthermal Ions Distribution

2008 ◽  
Vol 63 (5-6) ◽  
pp. 261-272 ◽  
Author(s):  
Hesham G. Abdelwahed ◽  
Emad K. El-Shewy ◽  
Mohsen A. Zahran ◽  
Mohamed T. Attia
Keyword(s):  
Dusty Plasma ◽  
Kdv Equation ◽  
Higher Order ◽  
Reductive Perturbation Method ◽  
Charged Dust ◽  
Ion Distributions ◽  
Kdv Equations ◽  
Dust Acoustic ◽  
Renormalization Method ◽  
Higher Order Corrections

Propagation of nonlinear dust-acoustic (DA) waves in a unmagnetized collisionless mesospheric dusty plasma containing positively and negatively charged dust grains and nonthermal ion distributions are investigated. For nonlinear DA waves, a reductive perturbation method is employed to obtain a Korteweg-de Vries (KdV) equation for the first-order potential. As it is well-known, KdV equations contain the lowest-order nonlinearity and dispersion, and consequently can be adopted for only small amplitudes. As the wave amplitude increases, the width and velocity of a soliton can not be described within the framework of KdV equations. So, we extend our analysis and take higher-order nonlinear and dispersion terms into account to clarify the essential effects of higher-order corrections. Moreover, in order to study the effects of higher-order nonlinearity and dispersion on the output solution, we address an appropriate technique, namely the renormalization method.

Integrable Models

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Fluid Mechanics ◽  
Soliton Solution ◽  
Water Waves ◽  
Heisenberg Model ◽  
Kdv Equation ◽  
Short Review ◽  
Lax Pair ◽  
Shallow Water Waves ◽  
Initial Value ◽  
The Kdv Equation

Some exceptional situations in fluid mechanics can be modeled by equations that are analytically solvable. The most famous example is the Korteweg–de Vries (KdV) equation for shallow water waves in a channel. The exact soliton solution of this equation is derived. The Lax pair formalism for solving the general initial value problem is outlined. Two hamiltonian formalisms for the KdV equation (Fadeev–Zakharov and Magri) are explained. Then a short review of the geometry of curves (Frenet–Serret equations) is given. They are used to derive a remarkably simple equation for the propagation of a kink along a vortex filament. This equation of Hasimoto has surprising connections to the nonlinear Schrödinger equation and to the Heisenberg model of ferromagnetism. An exact soliton solution is found.

scholarly journals Wave Breaking in Undular Bores with Shear Flows

Water Waves ◽  
2021 ◽  
Author(s):  
Maria Bjørnestad ◽  
Henrik Kalisch ◽  
Malek Abid ◽  
Christian Kharif ◽  
Mats Brun
Keyword(s):  
Turbulent Flows ◽  
Wave Breaking ◽  
Vries Equation ◽  
Kdv Equation ◽  
Flow Depth ◽  
Present Contribution ◽  
Undular Bore ◽  
Wave Flume ◽  
The Kdv Equation ◽  
The Waves

AbstractIt is well known that weak hydraulic jumps and bores develop a growing number of surface oscillations behind the bore front. Defining the bore strength as the ratio of the head of the undular bore to the undisturbed depth, it was found in the classic work of Favre (Ondes de Translation. Dunod, Paris, 1935) that the regime of laminar flow is demarcated from the regime of partially turbulent flows by a sharply defined value 0.281. This critical bore strength is characterized by the eventual breaking of the leading wave of the bore front. Compared to the flow depth in the wave flume, the waves developing behind the bore front are long and of small amplitude, and it can be shown that the situation can be described approximately using the well known Kortweg–de Vries equation. In the present contribution, it is shown that if a shear flow is incorporated into the KdV equation, and a kinematic breaking criterion is used to test whether the waves are spilling, then the critical bore strength can be found theoretically within an error of less than ten percent.

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