# American Institute of Mathematical Sciences

August  2018, 11(4): 691-705. doi: 10.3934/dcdss.2018043

## A study of bifurcation parameters in travelling wave solutions of a damped forced Korteweg de Vries-Kuramoto Sivashinsky type equation

 International Center for Applied Mathematics and Computational Bioengineering, Department of Mathematics and Natural Sciences, Gulf University for Science and Technology, Kuwait 300 College Park, Department of Mathematics, University of Dayton, Dayton, Ohio 45469-2316, USA Department of Mechanical Engineering, 300 College Park, University of Dayton, Dayton, Ohio 45469, USA

Authors name are in alphabetical order.

Received  January 2017 Revised  June 2017 Published  November 2017

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.

Citation: Mudassar Imran, Youssef Raffoul, Muhammad Usman, Chi Zhang. A study of bifurcation parameters in travelling wave solutions of a damped forced Korteweg de Vries-Kuramoto Sivashinsky type equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 691-705. doi: 10.3934/dcdss.2018043
##### References:

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##### References:
Three solutions corresponding to $f=2.5$
External Excitation Response Curve
Frequency-Response Curve without delay
Frequency-Response Curve with varying B
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