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The existence and decay of solitary waves for the Fornberg–Whitham equation

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Abstract

In this paper, we consider the Fornberg–Whitham equation and a family of solitary wave solutions is found by using minimization principle, where a related penalization function and the concentration-compactness lemma play a key role in the proof. Besides, we also prove that the family of solitary solutions is orbitally stable and decays exponentially when wave speed c is bigger than 1.

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References

  1. Fornberg, B., Whitham, G.B.: A numerical and theoretical study of certain nonlinear wave phenomena. Philos. Trans. R. Soc. Lond. Ser. A 289, 373–404 (1978)

    Article  MathSciNet  Google Scholar 

  2. Zhou, J., Tian, L.: A type of bounded traveling wave solutions for the Fornberg–Whitham equation. J. Math. Anal. Appl. 346, 255–261 (2008)

    Article  MathSciNet  Google Scholar 

  3. Yin, J., Tian, L., Fan, X.: Classification of travelling waves in the Fornberg–Whitham equation. J. Math. Anal. Appl. 368, 133–143 (2010)

    Article  MathSciNet  Google Scholar 

  4. Wu, X., Zhang, Z.: On the blow-up of solutions for the Fornberg–Whitham equation. Nonlinear Anal. Real World Appl. 44, 573–588 (2018)

    Article  MathSciNet  Google Scholar 

  5. Wei, L.: Wave breaking analysis for the Fornberg–Whitham equation. J. Differ. Equ. 265, 2886–2896 (2018)

    Article  MathSciNet  Google Scholar 

  6. Holmes, J.M.: Well-posedness of the Fornberg–Whitham equation on the circle. J. Differ. Equ. 260, 8530–8549 (2016)

    Article  MathSciNet  Google Scholar 

  7. Tian, L., Gao, Y.: The global attractor of the viscous Fornberg–Whitham equation. Nonlinear Anal. 71, 5176–5186 (2009)

    Article  MathSciNet  Google Scholar 

  8. Ehrnström, M., Wahlén, E.: On Whitham’s conjecture of a highest cusped wave for a nonlocal dispersive equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 36, 1603–1637 (2019)

    Article  MathSciNet  Google Scholar 

  9. Günther, H.: Discontinuous traveling waves as weak solutions to the Fornberg–Whitham equation. J. Differ. Equ. 265, 2825–2841 (2018)

    Article  MathSciNet  Google Scholar 

  10. Xu, F., Zhang, Y., Li, F.: The well-posedness, blow-up and travelling waves for a two-component Fornberg–Whitham system. J. Math. Phys. 62, 18 (2021)

    MATH  Google Scholar 

  11. Rossen, I.: On the integrability of a class of nonlinear dispersive wave equations. J. Nonlinear Math. Phys. 12, 462–468 (2005)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Chen, H., Bona, J.L.: Existence and asymptotic properties of solitary-wave solutions of Benjamin-type equations. Adv. Differ. Equ. 3, 51–84 (1998)

    MathSciNet  MATH  Google Scholar 

  14. Benjamin, T.B., Bona, J.L., Mahony, J.J.: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. Ser. A 272, 47–78 (1972)

    Article  MathSciNet  Google Scholar 

  15. Buffoni, B.: Existence and conditional energetic stability of capillary-gravity solitary water waves by minimisation. Arch. Ration. Mech. Anal. 173, 25–68 (2004)

    Article  MathSciNet  Google Scholar 

  16. Buffoni, B., Séré, E., Toland, J.F.: Minimization methods for quasi-linear problems with an application to periodic water waves. SIAM J. Math. Anal. 36, 1080–1094 (2005)

    Article  MathSciNet  Google Scholar 

  17. Breit, D., Wahlén, E.: A variational approach to solitary gravity-capillary interfacial waves with infinite depth. J. Nonlinear Sci. 29, 2601–2655 (2019)

    Article  MathSciNet  Google Scholar 

  18. Duchene, V., Nilsson, D., Wahlén, E.: Solitary wave solutions to a class of modified Green–Naghdi systems. J. Math. Fluid Mech. 20, 1059–1091 (2018)

    Article  MathSciNet  Google Scholar 

  19. Lions, P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case I. Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 109–145 (1984)

    Article  MathSciNet  Google Scholar 

  20. Bona, J.L., Li, Y.A.: Decay and analyticity of solitary waves. J. Math. Pures et Appliq. 76, 377–430 (1997)

    Article  MathSciNet  Google Scholar 

  21. Bruell, G., Ehrnström, M., Pei, L.: Symmery and decay of travelling wave solutions to the Whitham equation. J. Differ. Equ. 262, 4232–4254 (2017)

    Article  Google Scholar 

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Acknowledgements

This project is supported by National Natural Science Foundation of China (No: 11571057).

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  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

    Yong Zhang, Fei Xu & Fengquan Li

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  1. Yong Zhang
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  2. Fei Xu
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  3. Fengquan Li
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Correspondence to Fengquan Li.

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Zhang, Y., Xu, F. & Li, F. The existence and decay of solitary waves for the Fornberg–Whitham equation. Z. Angew. Math. Phys. 72, 112 (2021). https://doi.org/10.1007/s00033-021-01547-1

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  • DOI: https://doi.org/10.1007/s00033-021-01547-1

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