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Hybrid solitary wave solutions of the Camassa–Holm equation

  • Hugues M. Omanda , Clovis T. Djeumen Tchaho and Didier Belobo Belobo ORCID logo EMAIL logo
Published/Copyright: October 10, 2022
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International Journal of Nonlinear Sciences and Numerical Simulation
From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 24 Issue 5

Abstract

The Camassa–Holm equation governs the dynamics of shallow water waves or in its reduced form models nonlinear dispersive waves in hyperelastic rods. By using the straightforward Bogning-Djeumen Tchaho-Kofané method, explicit expressions of many solitary wave solutions with different profiles not previously derived in the literature are constructed and classified. Geometric characterizations of the solutions in terms of three new mappings are presented. Intensive numerical simulations carried confirm the stability of the solutions even with relatively high critical velocities and reveal that solitary waves with large widths are more stable than the ones with small widths.

Keywords: Camassa–Holm equation; hybrid solitary waves solutions; numerical simulations; traveling waves
Corresponding author: Didier Belobo Belobo, African Centre for Advanced Studies, P.O. Box 4477, Yaounde, Cameroon; Department of Mathematics and Physical Sciences, National Advanced School of Engineering, University of Yaounde I, P.O. Box 8390, Yaounde, Cameroon; and Laboratory of Biophysics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon, E-mail:

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: D. B. B. gratefully acknowledges support via a research fellowship from The World Academy of Sciences (TWAS) and the German Research Foundation (DFG) through the TWAS-DFG Cooperation Visits Programme.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix

The following equations stemming from Eq. (7) were used to derive the families of solutions presented above.

Term in sech11(αξ): (A.1) 510α 3 a 1 b 1 = 0

the term in sech10(αξ): (A.2) 336α 3 c 1 b 1 = 0

Term in sech9(αξ): (A.3) 204 α 3 ⁡ a + ( 24 α − 1037 α 3 ) a 1 b 1 − 296 α 3 a 1 b = 0

Term in sech8(αξ): (A4) − 108 α 3 c + ( 21 α − 862 α 3 ) c 1 b 1 − 182 α 3 c 1 b = 0

Term in sech7(αξ): (A5) ( 18 α − 363 α 3 ) a + ( 634 α 3 − 39 α ) a 1 b 1 + − 102 α 3 a + ( 409 α 3 − 18 α ) a 1 b = 0

Term in sech6(αξ): (A6) ( 170 α 3 − 15 α ) c + ( 696 α 3 − 63 α ) c 1 b 1 + 38 α 3 c + ( 336 α 3 − 15 α ) c 1 b = 0

Term in sech5(αξ): (A7) ( 198 α 3 − 27 α ) a + ( 15 α − 125 α 3 ) a 1 b 1 + ( 121 α 3 − 12 α ) a + ( 15 α − 125 α 3 ) a 1 b = 0

Term in sech4(αξ): (A8) 6 α k − ( 18 α 3 + 3 α ) ν + ( 30 α − 34 α 3 ) c + ( 63 α − 150 α 3 ) c 1 b 1 + 6 α 3 ν + ( 9 α − 46 α 3 ) c + ( 30 α − 174 α 3 ) c 1 b = 0

Term in sech3(αξ): (A9) (9α − 27α 3)(b 1 + b)a = 0

Term in sech2(αξ): (A10) ( 3 α + 36 α 3 ) ν − 6 α k − ( 28 α 3 + 15 α ) c − ( 20 α 3 + 21 α ) c 1 b 1 + ( α − 4 α 3 ) ν − 2 α k + ( 12 α 3 − 9 α ) c + ( 20 α 3 − 15 α ) c 1 b = 0 .

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Received: 2021-08-29
Revised: 2022-09-06
Accepted: 2022-09-18
Published Online: 2022-10-10

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/ijnsns-2021-0340
Keywords for this article
Camassa–Holm equation; hybrid solitary waves solutions; numerical simulations; traveling waves

Articles in the same Issue

  1. Frontmatter
  2. Original Research Article
  3. Hybrid solitary wave solutions of the Camassa–Holm equation
  4. Numerical simulations of wave propagation in a stochastic partial differential equation model for tumor–immune interactions
  5. A Chebyshev collocation method for solving the non-linear variable-order fractional Bagley–Torvik differential equation
  6. Higher order codimension bifurcations in a discrete-time toxic-phytoplankton–zooplankton model with Allee effect
  7. Numerical modeling of the dam-break flood over natural rivers on movable beds
  8. A class of piecewise fractional functional differential equations with impulsive
  9. Lie symmetry analysis for two-phase flow with mass transfer
  10. Asymptotic behavior for stochastic plate equations with memory in unbounded domains
  11. Discussion on controllability of non-densely defined Hilfer fractional neutral differential equations with finite delay
  12. A linearized finite difference scheme for time–space fractional nonlinear diffusion-wave equations with initial singularity
  13. Ground state solutions of Schrödinger system with fractional p-Laplacian
  14. Bifurcation analysis of a new stochastic traffic flow model
  15. An uncertainty measure based on Pearson correlation as well as a multiscale generalized Shannon-based entropy with financial market applications
  16. Hilfer fractional stochastic evolution equations on infinite interval
  17. Iterative learning control for conformable stochastic impulsive differential systems with randomly varying trial lengths
  18. Chebyshev wavelet-Picard technique for solving fractional nonlinear differential equations
  19. Theoretical assessment of the impact of awareness programs on cholera transmission dynamic
  20. Theoretical and numerical analysis of a prey–predator model (3-species) in the frame of generalized Mittag-Leffler law
  21. Controllability discussion for fractional stochastic Volterra–Fredholm integro-differential systems of order 1 < r < 2
  22. Shehu transform on time-fractional Schrödinger equations – an analytical approach
  23. A (2 + 1)-dimensional variable-coefficients extension of the Date–Jimbo–Kashiwara–Miwa equation: Lie symmetry analysis, optimal system and exact solutions
  24. Mathematical model of fluid flow in a double constricted tapered tube with permeable boundary
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