Dynamical behavior of traveling waves in a generalized VP-mVP equation with non-homogeneous power law nonlinearity

Feiting Fan; Xingwu Chen; Feiting Fan; Xingwu Chen

doi:10.3934/math.2023895

AIMS Mathematics

2023, Volume 8, Issue 8: 17514-17538. doi: 10.3934/math.2023895

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Research article

Dynamical behavior of traveling waves in a generalized VP-mVP equation with non-homogeneous power law nonlinearity

Feiting Fan ¹,
Xingwu Chen ^{2
,
,}

1.
School of Science, Civil Aviation Flight University of China, Guanghan, Sichuan 618307, China
2.
School of Mathematics, Sichuan University, Chengdu, Sichuan 610065, China

Received: 21 March 2023 Revised: 14 May 2023 Accepted: 16 May 2023 Published: 22 May 2023
MSC : 35C07, 35C08, 37G10, 37J46

In this paper, we investigate the dynamical behavior of traveling waves for a generalized Vakhnenko-Parkes-modified Vakhnenko-Parkes (VP-mVP) equation with non-homogeneous power law nonlinearity. By the dynamical systems approach and the singular traveling wave theory, the existence of all possible bounded traveling wave solutions is discussed, including smooth solutions (solitary wave solutions, periodic wave solutions and breaking wave solutions) and non-smooth solutions (solitary cusp wave solutions and periodic cusp wave solutions). We not only obtain all the explicit parametric conditions for the existence of 5 kinds of bounded traveling wave solutions, but also give their exact explicit expressions. Moreover, we qualitatively analyze the dynamical behavior of these traveling waves by using the bifurcation of phase portraits under different parameter conditions, and strictly prove the evolution of different traveling waves with their exact expressions.
- a generalized VP-mVP equation,
- traveling wave,
- existence,
- evolution,
- exact solution
Citation: Feiting Fan, Xingwu Chen. Dynamical behavior of traveling waves in a generalized VP-mVP equation with non-homogeneous power law nonlinearity[J]. AIMS Mathematics, 2023, 8(8): 17514-17538. doi: 10.3934/math.2023895

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Abstract

In this paper, we investigate the dynamical behavior of traveling waves for a generalized Vakhnenko-Parkes-modified Vakhnenko-Parkes (VP-mVP) equation with non-homogeneous power law nonlinearity. By the dynamical systems approach and the singular traveling wave theory, the existence of all possible bounded traveling wave solutions is discussed, including smooth solutions (solitary wave solutions, periodic wave solutions and breaking wave solutions) and non-smooth solutions (solitary cusp wave solutions and periodic cusp wave solutions). We not only obtain all the explicit parametric conditions for the existence of 5 kinds of bounded traveling wave solutions, but also give their exact explicit expressions. Moreover, we qualitatively analyze the dynamical behavior of these traveling waves by using the bifurcation of phase portraits under different parameter conditions, and strictly prove the evolution of different traveling waves with their exact expressions.

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