American Institute of Mathematical Sciences

Advanced
  • Previous Article
    The hypercircle theorem for elastic shells and the accuracy of Novozhilov's simplified equations for general cylindrical shells
  • DCDS-B Home
  • This Issue
  • Next Article
    Resonant oscillations of an inhomogeneous gas in a closed cylindrical tube
May  2007, 7(3): 629-641. doi: 10.3934/dcdsb.2007.7.629

Higher-order shallow water equations and the Camassa-Holm equation

David F. Parker 1,

1. 

School of Mathematics and Maxwell Institute for Mathematical Sciences, The King's Buildings, University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom

Received  June 2006 Revised  January 2007 Published  February 2007

The Korteweg-de Vries (KdV) equation has long been known to describe shallow water waves in an appropriate asymptotic limit. The Camassa-Holm (CH) equation, shown in 1993 [1] also to be completely integrable, allows solitons including peakons with discontinuous derivative. Consequently, its relevance to shallow water theory has been much doubted until the link was established [4] in 2004. Here, the perturbation procedure in terms of the standard amplitude ($\varepsilon$) and shallow depth ($\delta$) parameters shows that, while the KdV equation applies for $\delta^4 $«$ \varepsilon $«$ \delta$ with error term the larger of $\varepsilon^3$ and $\delta^6$,. However, the formal link to the CH equation imposes additional constraints on the coordinate ranges which are applicable. The derivation procedure is also extended to account for depth variations with bed slope $\mathcal{O}(\gamma\delta)$, provided that $\gamma $«$ \varepsilon$ and $\gamma $«$ \delta^2$ for $\delta^3 $«$ \varepsilon $«$ \delta$ and, within these parameter ranges, a theory for modulated waveforms is outlined. This utilizes expressions (in terms of elliptic functions) for the general travelling wave solutions to a (non-integrable) generalization of the CH equation. These include, as limiting cases, solitary waves with adjustable, but constrained, peak curvature (of $\mathcal{O}(\varepsilon \delta^2)$). The nonlinear dispersion relation for the periodic waveforms is illustrated and sample periodic waveforms are illustrated and compared to equivalent KdV approximations.
Keywords: Water waves, nonlinear dispersion., Camassa-Holm equation.
Mathematics Subject Classification: Primary: 35Q53, 76B15; Secondary: 35Q7.
Citation: David F. Parker. Higher-order shallow water equations and the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 629-641. doi: 10.3934/dcdsb.2007.7.629
[1]

Stephen C. Anco, Elena Recio, María L. Gandarias, María S. Bruzón. A nonlinear generalization of the Camassa-Holm equation with peakon solutions. Conference Publications, 2015, 2015 (special) : 29-37. doi: 10.3934/proc.2015.0029
[2]

Min Zhu, Shuanghu Zhang. Blow-up of solutions to the periodic modified Camassa-Holm equation with varying linear dispersion. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 7235-7256. doi: 10.3934/dcds.2016115
[3]

Min Zhu, Ying Wang. Blow-up of solutions to the periodic generalized modified Camassa-Holm equation with varying linear dispersion. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 645-661. doi: 10.3934/dcds.2017027
[4]

Delia Ionescu-Kruse. Variational derivation of the Camassa-Holm shallow water equation with non-zero vorticity. Discrete & Continuous Dynamical Systems, 2007, 19 (3) : 531-543. doi: 10.3934/dcds.2007.19.531
[5]

Yongsheng Mi, Boling Guo, Chunlai Mu. Persistence properties for the generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1623-1630. doi: 10.3934/dcdsb.2019243
[6]

Yu Gao, Jian-Guo Liu. The modified Camassa-Holm equation in Lagrangian coordinates. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2545-2592. doi: 10.3934/dcdsb.2018067
[7]

Yongsheng Mi, Boling Guo, Chunlai Mu. On an $N$-Component Camassa-Holm equation with peakons. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1575-1601. doi: 10.3934/dcds.2017065
[8]

Helge Holden, Xavier Raynaud. Dissipative solutions for the Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1047-1112. doi: 10.3934/dcds.2009.24.1047
[9]

Zhenhua Guo, Mina Jiang, Zhian Wang, Gao-Feng Zheng. Global weak solutions to the Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 883-906. doi: 10.3934/dcds.2008.21.883
[10]

Defu Chen, Yongsheng Li, Wei Yan. On the Cauchy problem for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 871-889. doi: 10.3934/dcds.2015.35.871
[11]

Milena Stanislavova, Atanas Stefanov. Attractors for the viscous Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 159-186. doi: 10.3934/dcds.2007.18.159
[12]

Aiyong Chen, Xinhui Lu. Orbital stability of elliptic periodic peakons for the modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1703-1735. doi: 10.3934/dcds.2020090
[13]

Li Yang, Zeng Rong, Shouming Zhou, Chunlai Mu. Uniqueness of conservative solutions to the generalized Camassa-Holm equation via characteristics. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5205-5220. doi: 10.3934/dcds.2018230
[14]

Shouming Zhou, Chunlai Mu. Global conservative and dissipative solutions of the generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1713-1739. doi: 10.3934/dcds.2013.33.1713
[15]

Yongsheng Mi, Chunlai Mu. On a three-Component Camassa-Holm equation with peakons. Kinetic & Related Models, 2014, 7 (2) : 305-339. doi: 10.3934/krm.2014.7.305
[16]

Shihui Zhu. Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 5201-5221. doi: 10.3934/dcds.2016026
[17]

Feng Wang, Fengquan Li, Zhijun Qiao. On the Cauchy problem for a higher-order μ-Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 4163-4187. doi: 10.3934/dcds.2018181
[18]

Danping Ding, Lixin Tian, Gang Xu. The study on solutions to Camassa-Holm equation with weak dissipation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 483-492. doi: 10.3934/cpaa.2006.5.483
[19]

Priscila Leal da Silva, Igor Leite Freire. An equation unifying both Camassa-Holm and Novikov equations. Conference Publications, 2015, 2015 (special) : 304-311. doi: 10.3934/proc.2015.0304
[20]

Stephen Anco, Daniel Kraus. Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4449-4465. doi: 10.3934/dcds.2018194

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences