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The Bifurcations and Exact Traveling Wave Solutions for a Nonlocal Hydrodynamic-Type System

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Abstract

A hydrodynamic-type system taking into account nonlocal effects is investigated. The exact traveling wave solutions including smooth solitary waves solutions, pseudo-peakons, periodic peakons, compactons, kink, and anti-kink wave solutions and so on are derived via the method of dynamical systems and the theory of singular traveling wave systems. It is worth pointing out that the uncountably infinitely many solitary wave solutions, kink, and anti-kink solutions are new solutions for the modeling system.

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Fig. 1

Parameter values: ν = 1,(β, γ, g) = (0.5,1,− 2)

Fig. 2

Parameter values: ν = 1,(β, γ, g) = (0.5,− 1,− 2)

Fig. 3

Parameter values: ν = 0,(β, γ, g) = (0.7,1,− 2)

Fig. 4

Parameter values: ν = 0,(β, γ, g) = (0.7,− 1,− 2)

Fig. 5

Parameter values: γ = 1, a (β, g, s) = (2,2,0.2), b (β, g, s) = (2,− 2,1.8), c (β, g, s) = (2,− 2.1,1.8)

Fig. 6

Parameter values: γ = − 1, a (β, g, s) = (2,2,0.2), b (β, g, s) = (2,− 2,1.8), c (β, g, s) = (2,− 2.1,1.8)

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References

  1. David C, Feng Z. Solitary waves in fluid media. Sharjah: Bentham Science; 2010.

    Google Scholar 

  2. Li J. Singular nonlinear traveling wave equations: bifurcations and exact solutions. Beijing: Science Press; 2013.

    Google Scholar 

  3. Yang J. 2010. Nonlinear waves in integrable and nonintegrable systems. SIAM, Philadelphia.

  4. Lenells J. Traveling wave solutions of the Camassa-Holm and Korteweg-de Vries equations. J. Nonlinear Math. Phys. 2004;11:508–520.

    Article  MathSciNet  Google Scholar 

  5. Lenells J. Traveling wave solutions of the Camassa-Holm equation. J Differ Equ 2005;217:393–430.

    Article  MathSciNet  Google Scholar 

  6. Lenells J. Classification of all travelling-wave solutions for some nonlinear dispersive equations. Philos Transact Math Phys Eng Sci 2007;365:2291–2298.

    MathSciNet  MATH  Google Scholar 

  7. Li J, Chen G. On a class of singular nonlinear traveling wave equations. Int J Bifurcat Chaos 2007;17:4049–4065.

    Article  MathSciNet  Google Scholar 

  8. Li J, Zhu W, Chen G. Understanding peakons, periodic peakons and compactons via a shallow water wave equation. Int J Bifurcat Chaos 2016; 26:1650207.

    Article  MathSciNet  Google Scholar 

  9. da Silva PL. Classification of bounded travelling wave solutions for the Dullin-Gottwald-Holm equation. J Math Anal Appl 2019;471:481–488.

    Article  MathSciNet  Google Scholar 

  10. Yang J. Classification of the solitary waves in coupled nonlinear Schrödinger equations. Physica D 1997;108:92–112.

    Article  MathSciNet  Google Scholar 

  11. Yang J. Classification of solitary wave bifurcations in generalized nonlinear Schrödinger equations. Stud Appl Math 2012;129:133–162.

    Article  MathSciNet  Google Scholar 

  12. Zhang L, Khalique C. Classification and bifurcations of a class of second-order ODEs and its application to nonlinear PDEs. Discret Cont Dyn. S 2018;11: 759–772.

    MathSciNet  MATH  Google Scholar 

  13. Vladimirov VA, Kutafina EV. On the localized invariant solutions of some non-local hydrodynamic-type models. Proc Inst Math NAS Ukraine 2004; 50:1510–1517.

    MathSciNet  MATH  Google Scholar 

  14. Vladimirov VA, Kutafina EV, Zorychta B. On the non-local hydrodynamic-type system and its soliton-like solutions. J Phys Math Theor 2012;45:085210.

    Article  MathSciNet  Google Scholar 

  15. Vladimirov VA, Maczka C, Sergyeyev A, Skurativskyi S. Stability and dynamical features of solitary wave solutions for a hydrodynamic-type system taking into account nonlocal effects. Commun Nonlinear Sci Numer Simul 2014;19: 1770–1782.

    Article  MathSciNet  Google Scholar 

  16. Vladimirov VA, Skurativskyi S. 2018. On the spectral stability of soliton-like solutions to a non-local hydrodynamic-type model arXiv:1807.08494v1 [nlin.PS].

  17. Shi J, Li J. Bifurcation approach to analysis of travelling waves in nonlocal hydrodynamic-type models. Abstr Appl Anal 2014;2014:1–12.

    MathSciNet  Google Scholar 

  18. Chen A, Zhu W, Qiao Z, Huang W. Algebraic traveling wave solutions of a nonlocal hydrodynamic-type model. Math Phys Anal Geom 2014;17: 465–482.

    Article  MathSciNet  Google Scholar 

  19. Byrd PF, Fridman MD. Handbook of elliptic integrals for engineers and scientists. Berlin: Springer; 1971.

    Book  Google Scholar 

Download references

Funding

This study was funded by the National Natural Science Foundation of China [grant numbers 61573004, 11371326].

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Correspondence to Yi Zhang.

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Liang, J., Zhang, Y. The Bifurcations and Exact Traveling Wave Solutions for a Nonlocal Hydrodynamic-Type System. J Dyn Control Syst 27, 645–659 (2021). https://doi.org/10.1007/s10883-020-09509-y

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  • DOI: https://doi.org/10.1007/s10883-020-09509-y

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