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Existence of Single-Peak Solitary Waves and Double-Peaks Solitary Wave of Gardner Equation with Kuramoto–Sivashinsky Perturbation

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Abstract

This paper concerns geometric study of single-peak solitary waves and double-peaks solitary wave of Gardner equation with Kuramoto–Sivashinsky perturbation. We first reduce the high-dimensional traveling wave system of the perturbed Gardner equation to the perturbed planar system through geometric singular perturbation theory. We then show the persistence of one homoclinic orbit, and the generation of a new homoclinic orbit by the Melnikov function method. Single-peak solitary waves and double-peaks solitary wave are newly found for the perturbed Gardner equation. The numerical simulations are performed to verify the theoretical results.

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Acknowledgements

This work is partially supported by the National Natural Science Foundation of China (12071162), the Natural Science Foundation of Fujian Province (No. 2021J01302) and the Fundamental Research Funds for the Central Universities (No. ZQN-802).

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Authors and Affiliations

  1. School of Mathematical Sciences, Huaqiao University, Quanzhou, 362021, People’s Republic of China

    Keqin Zhao & Zhenshu Wen

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  1. Keqin Zhao
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  2. Zhenshu Wen
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Contributions

KZ carried out the analytical studies and wrote the draft paper. ZW conceived the study and the overall manuscript design, and reviewed and revised the paper. All authors read and approved the final manuscript.

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Correspondence to Zhenshu Wen.

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Zhao, K., Wen, Z. Existence of Single-Peak Solitary Waves and Double-Peaks Solitary Wave of Gardner Equation with Kuramoto–Sivashinsky Perturbation. Qual. Theory Dyn. Syst. 22, 112 (2023). https://doi.org/10.1007/s12346-023-00811-1

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