Bifurcations of traveling wave solutions for the mixed Korteweg-de Vries equation

Hui Wang; Xue Wang; Hui Wang; Xue Wang

doi:10.3934/math.2024081

AIMS Mathematics

2024, Volume 9, Issue 1: 1652-1663. doi: 10.3934/math.2024081

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Research article

Bifurcations of traveling wave solutions for the mixed Korteweg-de Vries equation

Hui Wang ^,,
Xue Wang

College of Science, Henan University of Engineering, Zhengzhou, 451191, China

Received: 10 October 2023 Revised: 16 November 2023 Accepted: 22 November 2023 Published: 13 December 2023
MSC : 35C07, 35C08, 74J35

In this paper, the bifurcation theory of planar dynamical systems is employed to investigate the mixed Korteweg-de Vries (KdV) equation. Under different parameter conditions, the bifurcation curves and phase portraits of corresponding Hamiltonian system are given. Furthermore, many different types of exact traveling waves are obtained, which include hyperbolic function solution, triangular function solution, rational solution and doubly periodic solutions in terms of the Jacobian elliptic functions. Furthermore, as all parameters in the representations of exact solutions are free variables, the solutions obtained show more complex dynamical behaviors, and could be applicable to explain diversity in qualitative features of wave phenomena.
- planar dynamical systems,
- bifurcation,
- the mixed KdV equation,
- solitary wave solutions,
- periodic wave solutions
Citation: Hui Wang, Xue Wang. Bifurcations of traveling wave solutions for the mixed Korteweg-de Vries equation[J]. AIMS Mathematics, 2024, 9(1): 1652-1663. doi: 10.3934/math.2024081

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Abstract

In this paper, the bifurcation theory of planar dynamical systems is employed to investigate the mixed Korteweg-de Vries (KdV) equation. Under different parameter conditions, the bifurcation curves and phase portraits of corresponding Hamiltonian system are given. Furthermore, many different types of exact traveling waves are obtained, which include hyperbolic function solution, triangular function solution, rational solution and doubly periodic solutions in terms of the Jacobian elliptic functions. Furthermore, as all parameters in the representations of exact solutions are free variables, the solutions obtained show more complex dynamical behaviors, and could be applicable to explain diversity in qualitative features of wave phenomena.

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