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Traveling Wave Solutions for Two Perturbed Nonlinear Wave Equations with Distributed Delay

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Abstract

Traveling wave solutions are a class of invariant solutions which are critical for shallow water wave equations. In this paper, traveling wave solutions for two perturbed KP-MEW equations with a local delay convolution kernel are examined. The model equation is reduced to a planar near-Hamiltonian system via geometric singular perturbation theorem, and the qualitative properties of the corresponding unperturbed system are analyzed by using dynamical system approach. The persistence of the bounded traveling wave solutions for the perturbed KP-MEW equations with delay is investigated. By using a criterion for the monotonicity of ratio of two Abelian integrals and Melnikov’s method, the existence of kink (anti-kink) wave solutions and periodic wave solutions of the model equation are established. The result shows that the delayed KP-MEW equations with positive perturbation and the one with negative perturbation exhibit completely diverse dynamical properties. These new findings greatly enrich the understanding of dynamical properties of the traveling wave solutions of perturbed nonlinear wave equations with local delay convolution kernel. Numerical experiments further confirm and illustrate the theoretical results.

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Acknowledgements

The authors would like to thank the anonymous referees’ valuable suggestions.

Funding

This work is supported by the National Natural Science Foundation of China (No. 12172199, 12011530062).

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Authors and Affiliations

  1. Qingdao University of Science and Technology, Gaomi, 261500, Shandong, China

    Jundong Wang & Na Ma

  2. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, Shandong, China

    Jundong Wang & Lijun Zhang

  3. Zhejiang Sci-Tech University, School of Computer Science and Technology, Hangzhou, 3100018, Zhejiang, China

    Xuwen Huo

  4. Material Science, Innovation and Modelling Research Focus Area, North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho, 2735, South Africa

    Lijun Zhang & Chaudry Masood Khalique

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  1. Jundong Wang
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JW wrote the original draft. LZ and XH performed formal analysis and validation. NM and CMK examined English spelling & grammar. All authors reviewed the manuscript.

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Correspondence to Lijun Zhang.

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Wang, J., Zhang, L., Huo, X. et al. Traveling Wave Solutions for Two Perturbed Nonlinear Wave Equations with Distributed Delay. Qual. Theory Dyn. Syst. 23, 175 (2024). https://doi.org/10.1007/s12346-024-01035-7

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