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Periodic and solitary waves in a Korteweg–de Vries equation with delay

Part of: Representations of solutions Local and nonlocal bifurcation theory Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  07 September 2023

Qi Qiao ,
Shuling Yan  and
Xiang Zhang
Qi Qiao
Affiliation:
School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou, Jiangsu 215009, People's Republic of China (qiaoqijs@163.com)
Shuling Yan
Affiliation:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, People's Republic of China (shulingyan@jsnu.edu.cn)
Xiang Zhang
Affiliation:
School of Mathematical Sciences, MOE–LSC, and CMA-Shanghai, Shanghai Jiao Tong University, Shanghai 200240, People's Republic of China (xzhang@sjtu.edu.cn)
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Abstract

For a perturbed generalized Korteweg–de Vries equation with a distributed delay, we prove the existence of both periodic and solitary waves by using the geometric singular perturbation theory and the Melnikov method. We further obtain monotonicity and boundedness of the speed of the periodic wave with respect to the total energy of the unperturbed system. Finally, we establish a relation between the wave speed and the wavelength.

Keywords

Korteweg–de Vries equation with delayperiodic wavesolitary wavegeometric singular perturbation theoryAbelian integral

MSC classification

Primary: 35Q53: KdV-like equations (Korteweg-de Vries)
Secondary: 35C07: Traveling wave solutions 37G15: Bifurcations of limit cycles and periodic orbits

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 155 , Issue 1 , February 2025 , pp. 176 - 198
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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