Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-26T08:52:09.671Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access Rights & Permissions [Opens in a new window]

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
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 23
Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ai, S. and Albashaireh, R.. Traveling waves in spatial SIRS models. J. Dyn. Differ. Equ. 26 (2014), 143–164.CrossRefGoogle ScholarPubMed
Ashwin, P., Bartuccelli, M. V., Bridges, T. J. and Gourley, S. A.. Travelling fronts for the KPP equation with spatio-temporal delay. Z. Angew. Math. Phys. 53 (2002), 103–122.CrossRefGoogle Scholar
Cacciafesta, F. and Suzzoni, A. S.. Continuity of the flow of KdV with regard to the Wasserstein metrics and application to an invariant measure. J. Differ. Equ. 259 (2015), 1024–1067.CrossRefGoogle Scholar
Carr, J.. Applications of centre manifold theory. Applied Mathematical Sciences, vol. 35 (New York-Berlin: Springer-Verlag, 1981).CrossRefGoogle Scholar
Chen, A., Guo, L. and Deng, X.. Existence of solitary waves and periodic waves for a perturbed generalized BBM equation. J. Differ. Equ. 261 (2016), 5324–5349.CrossRefGoogle Scholar
Chu, J., Coron, J. M. and Shang, P.. Asymptotic stability of a nonlinear Korteweg-de Vries equation with critical lengths. J. Differ. Equ. 259 (2015), 4045–4085.CrossRefGoogle Scholar
Dauxois, T. and Peyrard, M.. Physics of solitons. Reprint of the 2006 edition. Revised translation of the 2004 French edition (Cambridge: Cambridge University Press, 2010).Google Scholar
Derks, G. and van Gils, S.. On the uniqueness of traveling waves in perturbed Korteweg-de Vries equations. Jpn. J. Ind. Appl. Math. 10 (1993), 413–430.CrossRefGoogle Scholar
Du, Z., Li, J. and Li, X.. The existence of solitary wave solutions of delayed Camassa-Holm via a geometric approach. J. Funct. Anal. 275 (2018), 988–1007.CrossRefGoogle Scholar
Du, Z. and Qiao, Q.. The dynamics of traveling waves for a nonlinear Belousov-Zhabotinskii system. J. Differ. Equ. 269 (2020), 7214–7230.CrossRefGoogle Scholar
Fenichel, N.. Persistence and smoothness of invariant manifolds for ows. Indiana Univ. Math. J. 21 (1971), 193–226.CrossRefGoogle Scholar
Fenichel, N.. Asymptotic stability with rate conditions. Indiana Univ. Math. J. 23 (1974), 1109–1137.CrossRefGoogle Scholar
Fenichel, N.. Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31 (1979), 53–98.CrossRefGoogle Scholar
Gardner, C. S., Greene, J. M., Kruskal, M. D. and Miura, R. M.. Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett. 19 (1967), 1095–1097.CrossRefGoogle Scholar
Goubet, O.. Analyticity of the global attractor for damped forced periodic Korteweg-de Vries equation. J. Differ. Equ. 264 (2018), 3052–3066.CrossRefGoogle Scholar
Gourley, S. A. and Ruan, S.. The diffusive Nicholson's blowflies equation with distributed delay. Proc. R. Soc. Edinburgh 130A (2000), 1275–1291.CrossRefGoogle Scholar
Hakkaev, S., Iliev, I. D. and Kirchev, K.. Stability of periodic traveling waves for complex modified Korteweg-de Vries equation. J. Differ. Equ. 248 (2010), 2608–2627.CrossRefGoogle Scholar
Hayashi, N. and Naumkin, P. I.. Final state problem for Korteweg-de Vries type equations. J. Math. Phys. 47 (2006), 123501.CrossRefGoogle Scholar
He, X-Z, Ruan, S. and Xia, H.. Global stability in chemostat-type equations with distributed delays. SIAM J. Math. Anal. 29 (1998), 681–696.CrossRefGoogle Scholar
Isaza, P. and León, C. A.. On optimal exponential decay properties of solutions to the Korteweg-de Vries equation. J. Differ. Equ. 263 (2017), 5189–5215.CrossRefGoogle Scholar
Jones, C. K. R. T.. Geometrical singular perturbation theory. Dynamical Systems, Lecture Notes in Mathematics, vol. 1609 (Berlin: Springer, 1995).CrossRefGoogle Scholar
Korteweg, D. J. and de Vries, G.. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. 39 (1895), 422–443.CrossRefGoogle Scholar
Li, J., Lu, K. and Bates, P. W.. Geometric singular perturbation theory with real noise. J. Differ. Equ. 259 (2015), 5137–5167.CrossRefGoogle Scholar
Lv, G. and Wang, M.. Existence, uniqueness and asymptotic behavior of traveling wave fronts for a vector disease model. Nonlinear Anal. Real World Appl. 11 (2010), 2035–2043.CrossRefGoogle Scholar
Mansour, M. B. A.. A geometric construction of traveling waves in a generalized nonlinear dispersive-dissipative equation. J. Geom. Phys. 69 (2013), 116–122.CrossRefGoogle Scholar
Martel, Y.. Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations. Am. J. Math. 127 (2005), 1103–1140.CrossRefGoogle Scholar
Mizumachi, T.. Large time asymptotics of solutions around solitary waves to the generalized Korteweg-de Vries equations. SIAM J. Math. Anal. 32 (2001), 1050–1080.CrossRefGoogle Scholar
Miura, R. M.. The Korteweg-de Vries equation: a survey of results. SIAM Rev. 18 (1976), 412–459.CrossRefGoogle Scholar
Ogawa, T.. Travelling wave solutions to a perturbed Korteweg-de Vries equations. Hiroshima Math. J. 24 (1994), 401–422.CrossRefGoogle Scholar
Pelinovsky, E., Talipova, T. and Soomere, T.. The structure of algebraic solitons and compactons in the generalized Korteweg-de Vries equation. Physica D 419 (2021), 132785.CrossRefGoogle Scholar
Shen, J. and Zhang, X.. Traveling pulses in a coupled FitzHugh-Nagumo equation. Physica D 418 (2021), 132848.CrossRefGoogle Scholar
Ruan, S. and Xiao, D.. Stability of steady states and existence of travelling waves in a vector-disease model. Proc. R. Soc. Edinburgh Sect. A 134 (2004), 991–1011.CrossRefGoogle Scholar
Xu, Y., Du, Z. and Wei, L.. Geometric singular perturbation method to the existence and asymptotic behavior of traveling waves for a generalized Burgers-KdV equation. Nonlinear Dyn. 83 (2016), 65–73.CrossRefGoogle Scholar
Yan, W., Liu, Z. and Liang, Y.. Existence of solitary waves and periodic waves to a perturbed generalized KdV equation. Math. Model. Anal. 19 (2014), 537–555.CrossRefGoogle Scholar
Zhao, Z. and Ge, W.. Traveling wave solutions for Schrödinger equation with distributed delay. Appl. Math. Model. 35 (2011), 675–687.CrossRefGoogle Scholar
Zhuang, K., Du, Z. and Lin, X.. Solitary waves solutions of singularly perturbed higher-order KdV equation via geometric singular perturbation method. Nonlinear Dyn. 80 (2015), 629–635.CrossRefGoogle Scholar