Dynamics on traveling wave solutions for Broer-Kaup equation with distributed delay

Minzhi Wei; Feiting Fan; Xinxin Liu; Minzhi Wei; Feiting Fan; Xinxin Liu

doi:10.3934/math.2026037

AIMS Mathematics

2026, Volume 11, Issue 1: 857-880. doi: 10.3934/math.2026037

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Research article

Dynamics on traveling wave solutions for Broer-Kaup equation with distributed delay

1.
School of Data Science and Artificial Intelligence, Wenzhou University of Technology, Wenzhou 325035, China
2.
School of Mathematics and Quantitative Economics, Guangxi University of Finance and Economics, Nanning 530003, China
3.
School of Science, Civil Aviation Flight University of China, Guanghan 618307, China

Received: 06 November 2025 Revised: 25 December 2025 Accepted: 05 January 2026 Published: 12 January 2026
MSC : 34C25, 34C60, 37C27

This paper investigates the traveling wave solutions of the Broer-Kaup equation with distributed delay. Using geometric singular perturbation theory and the Melnikov function method, a qualitative analysis on the traveling wave equation is carried out. By restricting the system to a locally invariant manifold, the delayed traveling wave equation is reduced to a near-Hamiltonian system. A translation transformation is applied to simplify the near-Hamiltonian planar system. Through involution mapping criterion, the monotonicity of the ratio of two Abelian integrals is established, which ensures that the Melnikov function possesses a unique simple zero. Based on Poincaré and heteroclinic bifurcation theory, the sufficient conditions on the persistence of periodic and kink (anti-kink) wave solutions are derived. Moreover, we present numeric simulations to illustrate the given results.
- delayed Broer-Kaup equation,
- bifurcation theory,
- involution mapping criterion,
- Melnikov function,
- traveling wave solution
Citation: Minzhi Wei, Feiting Fan, Xinxin Liu. Dynamics on traveling wave solutions for Broer-Kaup equation with distributed delay[J]. AIMS Mathematics, 2026, 11(1): 857-880. doi: 10.3934/math.2026037

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Abstract

This paper investigates the traveling wave solutions of the Broer-Kaup equation with distributed delay. Using geometric singular perturbation theory and the Melnikov function method, a qualitative analysis on the traveling wave equation is carried out. By restricting the system to a locally invariant manifold, the delayed traveling wave equation is reduced to a near-Hamiltonian system. A translation transformation is applied to simplify the near-Hamiltonian planar system. Through involution mapping criterion, the monotonicity of the ratio of two Abelian integrals is established, which ensures that the Melnikov function possesses a unique simple zero. Based on Poincaré and heteroclinic bifurcation theory, the sufficient conditions on the persistence of periodic and kink (anti-kink) wave solutions are derived. Moreover, we present numeric simulations to illustrate the given results.

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