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Lie symmetries and singularity analysis for generalized shallow-water equations

  • Andronikos Paliathanasis EMAIL logo
Published/Copyright: August 10, 2020
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International Journal of Nonlinear Sciences and Numerical Simulation
From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 21 Issue 7-8

Abstract

We perform a complete study by using the theory of invariant point transformations and the singularity analysis for the generalized Camassa-Holm (CH) equation and the generalized Benjamin-Bono-Mahoney (BBM) equation. From the Lie theory we find that the two equations are invariant under the same three-dimensional Lie algebra which is the same Lie algebra admitted by the CH equation. We determine the one-dimensional optimal system for the admitted Lie symmetries and we perform a complete classification of the similarity solutions for the two equations of our study. The reduced equations are studied by using the point symmetries or the singularity analysis. Finally, the singularity analysis is directly applied on the partial differential equations from where we infer that the generalized equations of our study pass the singularity test and are integrable.

Keywords: Benjamin-Bono-Mahoney; Camassa-Holm; invariants; Lie symmetries; shallow water
Corresponding author: Andronikos Paliathanasis, Institute of Systems Science, Durban University of Technology, PO Box 1334, Durban, 4000, Republic of South Africa, andInstituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia, 5090000, Chile, Email:

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

  2. Research funding: None declared.

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

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Received: 2019-05-20
Accepted: 2020-05-26
Published Online: 2020-08-10
Published in Print: 2020-11-18

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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  1. Frontmatter
  2. Original Research Articles
  3. ANFIS based system identification of underactuated systems
  4. The dynamical behavior of mixed type lump solutions on the (3 + 1)-dimensional generalized Kadomtsev–Petviashvili–Boussinesq equation
  5. A generalization of weighted companion of Ostrowski integral inequality for mappings of bounded variation
  6. Bright and dark optical solitons for the generalized variable coefficients nonlinear Schrödinger equation
  7. A multivariate spectral quasi-linearization method for the solution of (2+1) dimensional Burgers’ equations
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  10. On the behaviors of rough multilinear fractional integral and multi-sublinear fractional maximal operators both on product Lp and weighted Lp spaces
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  12. Lie symmetries and singularity analysis for generalized shallow-water equations
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  14. Nonlinear wave equation in an inhomogeneous medium from non-standard singular Lagrangians functional with two occurrences of integrals
  15. Lie symmetry analysis and similarity solutions for the Jimbo – Miwa equation and generalisations
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  18. Boundary shape functions methods for solving the nonlinear singularly perturbed problems with Robin boundary conditions
  19. Global exponential stability analysis of anti-periodic solutions of discontinuous bidirectional associative memory (BAM) neural networks with time-varying delays
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https://doi.org/10.1515/ijnsns-2019-0152
Keywords for this article
Benjamin-Bono-Mahoney; Camassa-Holm; invariants; Lie symmetries; shallow water

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. ANFIS based system identification of underactuated systems
  4. The dynamical behavior of mixed type lump solutions on the (3 + 1)-dimensional generalized Kadomtsev–Petviashvili–Boussinesq equation
  5. A generalization of weighted companion of Ostrowski integral inequality for mappings of bounded variation
  6. Bright and dark optical solitons for the generalized variable coefficients nonlinear Schrödinger equation
  7. A multivariate spectral quasi-linearization method for the solution of (2+1) dimensional Burgers’ equations
  8. Global dissipativity and exponential synchronization of mixed time-varying delays neural networks with discontinuous activations
  9. Integrodifference master equation describing actively growing blood vessels in angiogenesis
  10. On the behaviors of rough multilinear fractional integral and multi-sublinear fractional maximal operators both on product Lp and weighted Lp spaces
  11. Approximate controllability of nonlinear Hilfer fractional stochastic differential system with Rosenblatt process and Poisson jumps
  12. Lie symmetries and singularity analysis for generalized shallow-water equations
  13. Conditions for the local and global asymptotic stability of the time–fractional Degn–Harrison system
  14. Nonlinear wave equation in an inhomogeneous medium from non-standard singular Lagrangians functional with two occurrences of integrals
  15. Lie symmetry analysis and similarity solutions for the Jimbo – Miwa equation and generalisations
  16. The barotropic Rossby waves with topography on the earth’s δ-surface
  17. Synchronization control between discrete uncertain networks with different topologies
  18. Boundary shape functions methods for solving the nonlinear singularly perturbed problems with Robin boundary conditions
  19. Global exponential stability analysis of anti-periodic solutions of discontinuous bidirectional associative memory (BAM) neural networks with time-varying delays
  20. A parallel hybrid implementation of the 2D acoustic wave equation
  21. Approximate controllability of fractional stochastic evolution equations with nonlocal conditions
  22. Fractional (3+1)-dim Jimbo Miwa system: invariance properties, exact solutions, solitary pattern solutions and conservation laws
  23. Optical solitons and stability analysis for the generalized second-order nonlinear Schrödinger equation in an optical fiber
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