Exact explicit nonlinear wave solutions to a modified cKdV equation

Zhenshu Wen; Lijuan Shi; Zhenshu Wen; Lijuan Shi

doi:10.3934/math.2020314

AIMS Mathematics

2020, Volume 5, Issue 5: 4917-4930. doi: 10.3934/math.2020314

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Research article

Exact explicit nonlinear wave solutions to a modified cKdV equation

Zhenshu Wen ^,,
Lijuan Shi

Fujian Province University Key Laboratory of Computational Science, School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, PR China

Received: 03 December 2019 Accepted: 31 May 2020 Published: 03 June 2020
MSC : 34C60, 35B3, 35C07

In this paper, we study nonlinear wave solutions to a modified cKdV equation by exploiting Bifurcation method of Hamiltonian systems. We identify all possible bifurcation conditions and obtain the phase portraits of the system in different regions of the parametric space, through which, we obtain exact explicit nonlinear wave solutions, including solitary wave solutions, singular wave solutions, periodic singular wave solutions, and kink (antikink) wave solutions. Of particular interest is the appearance of the so-called V-shaped kink (antikink) wave solutions, W-shaped solitary wave solutions, and W-shaped periodic wave solutions, which were not found in previous studies.
- modified cKdV equation,
- bifurcation,
- V-shaped kink (antikink) wave solutions,
- W-shaped solitary wave solutions,
- W-shaped periodic wave solutions
Citation: Zhenshu Wen, Lijuan Shi. Exact explicit nonlinear wave solutions to a modified cKdV equation[J]. AIMS Mathematics, 2020, 5(5): 4917-4930. doi: 10.3934/math.2020314

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Abstract

In this paper, we study nonlinear wave solutions to a modified cKdV equation by exploiting Bifurcation method of Hamiltonian systems. We identify all possible bifurcation conditions and obtain the phase portraits of the system in different regions of the parametric space, through which, we obtain exact explicit nonlinear wave solutions, including solitary wave solutions, singular wave solutions, periodic singular wave solutions, and kink (antikink) wave solutions. Of particular interest is the appearance of the so-called V-shaped kink (antikink) wave solutions, W-shaped solitary wave solutions, and W-shaped periodic wave solutions, which were not found in previous studies.

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