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Orbital stability of solitary wave solutions of Kudryashov–Sinelshchikov equation

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Abstract

We study orbital stability of solitary wave solutions of the Kudryashov–Sinelshchikov equation in describing the pressure waves in a mixture liquid. Our approach is based on the theories of orbital stability presented by Grillakis, Shatah and Strauss, and relies on the sign of scalar function. By analyzing and proving conditions for stability or instability of solitary wave solution to the Kudryashov–Sinelshchikov equation in terms of parameters, the conditions in stability of solitary wave solution to the Kudryashov–Sinelshchikov equation are given.

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References

  1. N.A. Kudryashov, D.I. Sinelshchikov, Nonlinear waves in bubbly liquids with consideration for viscosity and heat transfer. Phys. Lett. A 374, 2011–2016 (2010)

    Article  ADS  Google Scholar 

  2. N.A. Kudryashov, D.I. Sinelshchikov, Nonlinear evolution equation for describing waves in bubbly liquids with viscosity and heat transfer. Appl. Math. Comput. 217, 414–421 (2010)

    MathSciNet  MATH  Google Scholar 

  3. G.M. Coclite, L. di Ruvo, A singular limit problem for the Kudryashov–Sinelshchikov. Z. Angew. Math. Mech. 97(9), 1020–1033 (2017)

    Article  MathSciNet  Google Scholar 

  4. N.A. Kudryashov, D.I. Sinelshchikov, Nonlinear waves in liquids with gas bubbles with account of viscosity and heat transfer. Fluid Dyn. 45, 96–112 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  5. M. Randrüüt, On the Kudryashov–Sinelshchikov equation for waves in bubble liquids. Phys. Lett. A 375, 3687–3692 (2011)

    Article  ADS  Google Scholar 

  6. B. He, Q. Meng, Y. Long, the Bifurcation and exact peakons, solitary and periodic wave solutions for the Kudryashov–Sinelshchikov equation. Commun. Nonlinear Sci. Numer. Simul. 17, 4137–4148 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  7. M. Nadhafikhah, V. Shurvani-Sh, Lie symmetry analysis of Kudryashov–Sinelshchikov equation. Math. Probl. Eng. 2011, 457697 (2011)

    MathSciNet  MATH  Google Scholar 

  8. Y. He, New Jacobi elliptic function solutions for the Kudryashov–Sinelshchikov equation using improved F-expansion method. Math. Probl. Eng. 2013, 104894 (2013)

    MathSciNet  MATH  Google Scholar 

  9. M. Randrüüt, M. Braun, On idential traveling-wave solutions of the Kudryashov–Sinelshchikov equation and related equations. Int. J. Nonlinear Mech. 58, 206–211 (2014)

    Article  ADS  Google Scholar 

  10. M. Randrüüt, M. Braun, Cnoidal eaves governed by the Kudryashov–Sinelshchikov equation. Phys. Lett. A 377, 1868–1874 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  11. S. Hakkaev, K. Kirchev, Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa–Holm equation. Commuun. Partial Differ. Equ. 30, 761–781 (2005)

    Article  MathSciNet  Google Scholar 

  12. S. Levandosky, Y. Liu, Stability of solitary waves of a generalized Ostrovsky equation. SIAM. J. Math. Anal. 38, 985–1011 (2006)

    Article  MathSciNet  Google Scholar 

  13. M. Grillakis, J. Shatah, W. Strauss, Stability theory of solitary waves in the presence of symmetry I. J. Funct. Anal. 74, 160–197 (1987)

    Article  MathSciNet  Google Scholar 

  14. J.L. Bona, P.E. Souganidis, W.A. Strauss, Stability and instability of solitary waves of Korteweg–de Vries type. Proc. R. Soc. Lond. A 411, 395–412 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  15. X.H. Liu, W.G. Zhang, Z.M. Li, The orbital stability of the solitary wave solutions of the generalized Camassa–Holm equation. J. Math. Anal. Appl. 398, 776–784 (2013)

    Article  MathSciNet  Google Scholar 

  16. P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part 1 and part 2. Ann. Inst. H. Poincare Anal. Non Lineaire 145, 223–283 (1984)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. School of Data-Science and Information-Engineering, Guizhou Minzu University, Guiyang, 550025, Guizhou, People’s Republic of China

    XiaoHua Liu

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  1. XiaoHua Liu
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Correspondence to XiaoHua Liu.

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The author declares that I have no conflict of interest.

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This project is supported by Fund of Science and Technology Department of GuiZhou ([2019]1162).

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Liu, X. Orbital stability of solitary wave solutions of Kudryashov–Sinelshchikov equation. Eur. Phys. J. Plus 135, 804 (2020). https://doi.org/10.1140/epjp/s13360-020-00821-x

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  • DOI: https://doi.org/10.1140/epjp/s13360-020-00821-x

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