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

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

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

scholarly journals The stability of solitary waves

1972 ◽  
Vol 328 (1573) ◽  
pp. 153-183 ◽  
Keyword(s):  
Solitary Wave ◽  
Spectral Theory ◽  
Solitary Waves ◽  
Vries Equation ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
De Vries ◽  
Long Time ◽  
Stability Of Solitary Waves ◽  
The Stability

The Korteweg-de Vries equation, which describes the unidirectional propagation of long waves in a wide class of nonlinear dispersive systems, is well known to have solutions representing solitary waves. The present analysis establishes that these solutions are stable, confirming a property that has for a long time been presumed. The demonstration of stability hinges on two nonlinear functionals which for solutions of the Korteweg-de Vries equation are invariant with time: these are introduced in § 2, where it is recalled that Boussinesq recognized their significance in relation to the stability of solitary waves. The principles upon which the stability theory is based are explained in § 3, being supported by a few elementary ideas from functional analysis. A proof that solitary wave solutions are stable is completed in § 4, the most exacting steps of which are accomplished by means of spectral theory. In appendix A a method deriving from the calculus of variations is presented, whereby results needed for the proof of stability may be obtained independently of spectral theory as used in § 4. In appendix B it is shown how the stability analysis may readily be adapted to solitary-wave solutions of the ‘regularized long-wave equation’ that has recently been advocated by Benjamin, Bona & Mahony as an alternative to the Korteweg-de Vries equation. In appendix C a variational principle is demonstrated relating to the exact boundaryvalue problem for solitary waves in water: this is a counterpart to a principle used in the present work (introduced in §2) and offers some prospect of proving the stability of exact solitary waves.

Optical and singular solitary waves to the PNLSE with third order dispersion in Kerr media via two integration approaches

Optik ◽  
2018 ◽  
Vol 163 ◽  
pp. 142-151 ◽  
Author(s):  
Mustafa Inc ◽  
Aliyu Isa Aliyu ◽  
Abdullahi Yusuf ◽  
Dumitru Baleanu
Keyword(s):  
Solitary Waves ◽  
Third Order ◽  
Kerr Media ◽  
Third Order Dispersion ◽  
Order Dispersion

New Solitary-Wave Solutions for the Generalized Reaction Duffing Model and their Dynamics

2004 ◽  
Vol 59 (11) ◽  
pp. 721-728 ◽  
Author(s):  
Jong-Jae Kim ◽  
Woo-Pyo Hong
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Driving Force ◽  
Function Method ◽  
Auxiliary Function ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Dynamical Properties ◽  
Duffing Model ◽  
52.35 Mw

We find new analytic solitary-wave solutions, having a nonzero background at infinity, of the generalized reaction Duffing model 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 nonlinear terms may act as an effective driving force. - PACS numbers: 03.40.Kf, 02.30.Jr, 47.20.Ky, 52.35.Mw

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