Existence of traveling waves in a delayed convecting shallow water fluid model

Minzhi Wei; Minzhi Wei

doi:10.3934/era.2023343

Electronic Research Archive

2023, Volume 31, Issue 11: 6803-6819. doi: 10.3934/era.2023343

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Research article

Existence of traveling waves in a delayed convecting shallow water fluid model

Minzhi Wei ^{1,2
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1.
School of Mathematics and Quantitative Economics, Guangxi University of Finance and Economics, Nanning, Guangxi 530003, China
2.
School of Mathematics, Sichuan University, Chengdu, Sichuan 610065, China

Received: 26 June 2023 Revised: 11 September 2023 Accepted: 27 September 2023 Published: 19 October 2023

This paper investigates a delayed shallow water fluid model that has not been studied in previous literature. Applying geometric singular perturbation theory, we prove the existence of traveling wave solutions for the model with a nonlocal weak delay kernel and local strong delay convolution kernel, respectively. When the convection term contains a nonlocal weak generic delay kernel, the desired heteroclinic orbit is obtained by using Fredholm theory and linear chain trick to prove the existence of two kink wave solutions under certain parametric conditions. When the model contains local strong delay convolution kernel and weak backward diffusion, under the same parametric conditions to the previous case, the corresponding traveling wave system can be reduced to a near-Hamiltonian system. The existence of a unique periodic wave solution is established by proving the uniqueness of zero of the Melnikov function. Uniqueness is proved by utilizing the monotonicity of the ratio of two Abelian integrals.
- shallow water fluid model,
- geometric singular perturbation theory,
- Abelian integral,
- Fredholm theory,
- traveling waves
Citation: Minzhi Wei. Existence of traveling waves in a delayed convecting shallow water fluid model[J]. Electronic Research Archive, 2023, 31(11): 6803-6819. doi: 10.3934/era.2023343

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Abstract

This paper investigates a delayed shallow water fluid model that has not been studied in previous literature. Applying geometric singular perturbation theory, we prove the existence of traveling wave solutions for the model with a nonlocal weak delay kernel and local strong delay convolution kernel, respectively. When the convection term contains a nonlocal weak generic delay kernel, the desired heteroclinic orbit is obtained by using Fredholm theory and linear chain trick to prove the existence of two kink wave solutions under certain parametric conditions. When the model contains local strong delay convolution kernel and weak backward diffusion, under the same parametric conditions to the previous case, the corresponding traveling wave system can be reduced to a near-Hamiltonian system. The existence of a unique periodic wave solution is established by proving the uniqueness of zero of the Melnikov function. Uniqueness is proved by utilizing the monotonicity of the ratio of two Abelian integrals.

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