Dynamics of Solitary-Waves in the Higher Order Korteweg –De Vries Equation Type (I)

2005 ◽  
Vol 60 (11-12) ◽  
pp. 757-767 ◽  
Author(s):  
Woo-Pyo Hong
Keyword(s):  
Solitary Waves ◽  
Wave Equations ◽  
Analytic Solution ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Auxiliary Function ◽  
Higher Order ◽  
Type I ◽  
De Vries ◽  
52.35 Mw

We find new analytic solitary-wave solutions of the higher order wave equations of Korteweg - De Vries (KdV) type (I), using the auxiliary function method. We study the dynamical properties of the solitary-waves by numerical simulations. It is shown that the solitary-waves are stable for wide ranges of the model coefficients. We study the dynamics of the two solitary-waves by using the analytic solution as initial profiles and find that they interact elastically in the sense that the mass and energy of the system are conserved. This leads to the possibility of multi-soliton solutions of the higher order KdV type (I), which can not be found by current analytical methods. - PACS numbers: 03.40.Kf, 02.30.Jr, 47.20.Ky, 52.35.Mw

Dynamics of Solitary-waves in the Coupled Korteweg-De Vries Equations

2005 ◽  
Vol 60 (8-9) ◽  
pp. 557-565
Author(s):  
Woo-Pyo Hong ◽  
Jong-Jae Kim
Keyword(s):  
Solitary Waves ◽  
Vries Equation ◽  
Auxiliary Function ◽  
Solitary Wave Solutions ◽  
Third Order ◽  
Third Order Dispersion ◽  
De Vries ◽  
Order Dispersion ◽  
Dispersion Term ◽  
52.35 Mw

We find new analytic solitary-wave solutions, having a nonzero background at infinity, of the coupled Korteweg-De Vries equation, using the auxiliary function method. We study the dynamical properties of the solitary-waves by numerical simulations. It is shown that the solitary-waves can be stable or unstable, depending on the coefficients of the model. We study the interaction dynamics by using the solitary-waves as initial profiles to show that the mass and energy of the coupled Korteweg- De Vries can be conserved for a negative third-order dispersion term. - PACS numbers: 03.40.Kf, 02.30.Jr, 47.20.Ky, 52.35.Mw

Noncommutative coupled complex modified Korteweg–de Vries equation: Darboux and binary Darboux transformations

2019 ◽  
Vol 34 (07n08) ◽  
pp. 1950054
Author(s):  
H. Wajahat A. Riaz
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Higher Order ◽  
Nonlinear Evolution Equations ◽  
Order Effects ◽  
Darboux Transformations ◽  
De Vries ◽  
Higher Order Effects

Higher-order nonlinear evolution equations are important for describing the wave propagation of second- and higher-order number of fields in optical fiber systems with higher-order effects. One of these equations is the coupled complex modified Korteweg–de Vries (ccmKdV) equation. In this paper, we study noncommutative (nc) generalization of ccmKdV equation. We present Darboux and binary Darboux transformations (DTs) for the nc-ccmKdV equation and then construct its Quasi-Grammian solutions. Further, single and double-hump soliton solutions of first- and second-order are given in commutative settings.

scholarly journals Higher-order Korteweg-de Vries models for internal solitary waves in a stratified shear flow with a free surface

2002 ◽  
Vol 9 (3/4) ◽  
pp. 221-235 ◽  
Author(s):  
R. Grimshaw ◽  
E. Pelinovsky ◽  
O. Poloukhina
Keyword(s):  
Free Surface ◽  
Shear Flow ◽  
Solitary Waves ◽  
Vries Equation ◽  
Wave Theory ◽  
Higher Order ◽  
Internal Solitary Waves ◽  
Long Wave ◽  
De Vries ◽  
Stratified Shear Flow

Abstract. A higher-order extension of the familiar Korteweg-de Vries equation is derived for internal solitary waves in a density- and current-stratified shear flow with a free surface. All coefficients of this extended Korteweg-de Vries equation are expressed in terms of integrals of the modal function for the linear long-wave theory. An illustrative example of a two-layer shear flow is considered, for which we discuss the parameter dependence of the coefficients in the extended Korteweg-de Vries equation.

Non-integrable lattice equations supporting kink and soliton solutions

2001 ◽  
Vol 12 (6) ◽  
pp. 709-718 ◽  
Author(s):  
A. K. COMMON ◽  
M. MUSETTE
Keyword(s):  
Solitary Waves ◽  
Difference Equations ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Elastic Collision ◽  
Discrete Form ◽  
Integrable Lattice ◽  
De Vries ◽  
Burgers Hierarchy

Nonintegrable differential-difference equations are constructed which support two-kink and two-soliton solutions. These equations are related to the discrete Burgers hierarchy and a discrete form of the Korteweg-de Vries equation. In particular, discretisations of equations related to the Fitzhugh-Nagumo-Kolmogorov-Petrovskii-Piskunov, Satsuma-Burgers-Huxley equations are derived. Methods presented here can also be used to derive non-integrable differential-difference equations describing the elastic collision of more than two kinks or solitary waves.

The extended Korteweg-de Vries equation and the resonant flow of a fluid over topography

1990 ◽  
Vol 221 ◽  
pp. 263-287 ◽  
Author(s):  
T. R. Marchant ◽  
N. F. Smyth
Keyword(s):  
Water Wave ◽  
Wave Equations ◽  
Vries Equation ◽  
Phase Speed ◽  
Simple Wave ◽  
Flow Speed ◽  
Higher Order ◽  
Modulation Equations ◽  
Long Wave ◽  
De Vries

The extended Korteweg-de Vries equation which includes nonlinear and dispersive terms cubic in the wave amplitude is derived from the water-wave equations and the Lagrangian for the water-wave equations. For the special case in which only the higher-order nonlinear term is retained, the extended Korteweg-de Vries equation is transformed into the Korteweg-de Vries equation. Modulation equations for this equation are then derived from the modulation equations for the Korteweg-de Vries equation and the undular bore solution of the extended Korteweg-de Vries equation is found as a simple wave solution of these modulation equations. The modulation equations are also used to extend the solution for the resonant flow of a fluid over topography. This resonant flow occurs when, in the weakly nonlinear, long-wave limit, the basic flow speed is close to a linear long-wave phase speed for one of the long-wave modes. In addition to the effect of higher-order terms, the effect of boundary-layer viscosity is also considered. These solutions (with and without viscosity) are compared with recent experimental and numerical results.

On Solitary-Wave Solutions for the Coupled Korteweg – de Vries and Modified Korteweg – de Vries Equations and their Dynamics

2006 ◽  
Vol 61 (3-4) ◽  
pp. 125-132 ◽  
Author(s):  
Woo-Pyo Hong
Keyword(s):  
Solitary Wave ◽  
Numerical Simulations ◽  
Solitary Waves ◽  
Function Method ◽  
Auxiliary Function ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
De Vries ◽  
Dynamical Properties ◽  
52.35 Mw

Analytic sech4-type traveling solitary-wave solutions of the coupled Korteweg-de Vries and modified Korteweg-de Vries equations proposed by Kersten-Krasil’shchik, are found by applying the auxiliary function method. The dynamical properties of the solitary-waves are studied by numerical simulations. - PACS numbers: 03.40.Kf, 02.30.Jr, 47.20.Ky, 52.35.Mw

scholarly journals Multiple-Pole Solutions to a Semidiscrete Modified Korteweg-de Vries Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Zhixing Xiao ◽  
Kang Li ◽  
Junyi Zhu
Keyword(s):  
Soliton Solution ◽  
Vries Equation ◽  
Hilbert Problem ◽  
Soliton Solutions ◽  
Higher Order ◽  
Symmetry Condition ◽  
De Vries ◽  
Riemann Hilbert Problem

Multiple-pole soliton solutions to a semidiscrete modified Korteweg-de Vries equation are derived by virtue of the Riemann-Hilbert problem with higher-order zeros. A different symmetry condition is introduced to build the nonregular Riemann-Hilbert problem. The simplest multiple-pole soliton solution is presented. The dynamics of the solitons are studied.

Stability and instability of some nonlinear dispersive solitary waves in higher dimension

1996 ◽  
Vol 126 (1) ◽  
pp. 89-112 ◽  
Author(s):  
Anne de Bouard
Keyword(s):  
Solitary Waves ◽  
Acoustic Waves ◽  
Vries Equation ◽  
Three Dimensional ◽  
Higher Dimension ◽  
Ion Acoustic Waves ◽  
Stability And Instability ◽  
Ion Acoustic ◽  
De Vries ◽  
The Stability

We study the stability of positive radially symmetric solitary waves for a three dimensional generalisation of the Korteweg de Vries equation, which describes nonlinear ion-acoustic waves in a magnetised plasma, and for a generalisation in dimension two of the Benjamin–Bona–Mahony equation.

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