Abstract
In this work, we consider an ordinary differential equation obtained from a damped externally excited Korteweg de Vries-Kuramoto Sivashinsky (KdV-KS) type equation using traveling coordinates. We also include controls and delays and use an asymptotic perturbation method to analyze the stability of the traveling wave solutions. The existence of bounded solutions is presented as well. We consider the primary resonance defined by the detuning parameter. External-excitation and frequency-response curves are shown to exhibit jump and hysteresis phenomena (discontinuous transitions between two stable solutions) for the KdV-KS type equation. We have obtained the existence of the bounded solutions of the system obtained from an ordinary differential equation associated with the KdV-KS equation and also show the global stability for a special case when there is no external force.
Highlights
The Korteweg-de Vries (KdV) equation has a very interesting history going back to 1834, when a naval engineer Scott Russell observed a solitary wave [38]; the associated KdV equation for solitary waves was derived in 1895 by D
In [10], authors included damping and dissipation terms that make the associated differential equation irreducible to lower order and extended the asymptotic perturbation methods to study the bifurcation in steady states and studied the Benjamin-Bona-Mahony equation (BBM equation) [1]
We have considered a forced perturbed Korteweg de Vries-Kuramoto Sivashinsky (KdV-KS) equation with controls and delays
Summary
The Korteweg-de Vries (KdV) equation has a very interesting history going back to 1834, when a naval engineer Scott Russell observed a solitary wave (he called it a great wave of translation) [38]; the associated KdV equation for solitary waves was derived in 1895 by D. Korteweg de Vries equation, Kuramoto-Sivashinsky equation, KdV-KS equation, KS-type equation, bifurcations, steady state solutions, asymptotic perturbation method. The effect of dissipation, damping, dispersion and external forcing play an important role in the study of bifurcations in traveling wave solutions of third and fourth order partial differential equations. The traveling wave solutions of the Burgers-KdV equation with a fourth order term is studied by Mansour in [32].
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