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New solitary wave solutions of a generalized BBM equation with distributed delays

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Abstract

Solitary wave solutions for a generalized Benjamin–Bona–Mahony equation with distributed delay and dissipative perturbation are considered in this paper. The corresponding traveling wave equation is transformed into a four-dimensional dynamical system, which is regarded as a singularly perturbed system for small time delay. The four-dimensional dynamical system is reduced to a near-Hamiltonian planar system via geometric singular perturbation method. The existence of solitary wave solutions with a single crest or trough is established by proving the persistence of homoclinic orbits of the near-Hamiltonian system. More importantly, a new type of solitary wave solution with coexisting crest and trough, which corresponds to a large concave homoclinic orbit, is observed theoretically by using Melnikov’s method. The selection principle for wave speed of the solitary wave is presented which can be utilized directly to determine the limit wave speed. Numerical simulations are in complete agreement with the theoretical predictions.

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Funding

This work is partially supported by the National Natural Science Foundation of China No. 12172199 and No. 12011530062.

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Authors and Affiliations

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

    Jundong Wang, Lijun Zhang & Jibin Li

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

    Jundong Wang

  3. International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Mmabatho, South Africa

    Lijun Zhang

Authors
  1. Jundong Wang
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  2. Lijun Zhang
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  3. Jibin Li
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Contributions

LZ Validation, Methodology, Formal analysis, Writing - review. JW software, Formal analysis, Writing - original draft & editing. JL validation, Methodology.

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Correspondence to Lijun Zhang.

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Wang, J., Zhang, L. & Li, J. New solitary wave solutions of a generalized BBM equation with distributed delays. Nonlinear Dyn 111, 4631–4643 (2023). https://doi.org/10.1007/s11071-022-08043-9

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  • DOI: https://doi.org/10.1007/s11071-022-08043-9

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