Abstract
This paper is devoted to studying wave breaking phenomena for the Fornberg–Whitham equation. Making use of the structure and the conservation quantity of the Fornberg–Whitham equation, we give a sufficient condition on the initial data to ensure wave breaking.
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Whitham, G.B.: Variational methods and applications to water waves. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 299(1456), 6–25 (1967)
Fornberg, B.: A numerical and theoretical study of certain nonlinear wave phenomena. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci. 289(1361), 373–404 (1978)
Naumkin, P.I., Shishmarev, I.A.: Nonlinear Nonlocal Equations in the Theory of Waves. Americn Mathematical Society, Providence (1994)
Holmes, J.M.: Well-posedness of the Fornberg–Whitham equation on the circle. J. Differ. Equ. 260(12), 8530–8549 (2016)
Holmes, J., Thompson, R.C.: Well-posedness and continuity properties of the Fornberg–Whitham equation in Besov spaces. J. Differ. Equ. 263(7), 4355–4381 (2017)
Haziot, S.V.: Wave breaking for the Fornberg–Whitham equation. J. Differ. Equ. 263(12), 8178–8185 (2017)
Hörmann, G.: Wave breaking of periodic solutions to the Fornberg–Whitham equation. Discrete Contin. Dyn. Syst. 38(3), 1605–1613 (2018)
Wei, L.: Wave breaking analysis for the Fornberg–Whitham equation. J. Differ. Equ. 265(7), 2886–2896 (2018)
Wu, X., Zhang, Z.: On the blow-up of solutions for the Fornberg–Whitham equation. Nonlinear Anal. Real World Appl. 44, 573–588 (2018)
Hörmann, G.: Discontinuous traveling waves as weak solutions to the Fornberg–Whitham equation. J. Differ. Equ. 265(7), 2825–2841 (2018)
Korteweg, D.J., Vries, D.: On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. 39, 422–443 (1895)
Drazin, P.G., Johnson, R.S.: Solitons: An Introduction. Cambridge University Press, Cambridge (1989)
Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71(11), 1661 (1993)
Camassa, R., Holm, D.D., Hyman, J.M.: A new integrable shallow water equation. Adv. Appl. Mech. 31, 1–33 (1994)
Dai, H.H.: Model equations for nonlinear dispersive waves in a compressible Mooney–Rivlin rod. Acta Mechanica 127(1–4), 193–207 (1998)
Fuchssteiner, B., Fokas, A.S.: Symplectic structures, their Bäklund transformations and hereditary symmetries. Physica D Nonlinear Phenom. 4(1), 47–66 (1981)
Constantin, A.: On the scattering problem for the Camassa–Holm equation. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 457(2008), 953–970 (2001)
Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Mathematica 181(2), 229–243 (1998)
Constantin, A., Escher, J.: Global existence and blow-up for a shallow water equation. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 26(2), 303–328 (1998)
Constantin, A.: Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Annales de l’institut Fourier 50(2), 321–362 (2000)
Constantin, A., Escher, J.: On the blow-up rate and the blow-up set of breaking waves for a shallow water equation. Mathematische Z. 233(1), 75–91 (2000)
Constantin, A.: The trajectories of particles in Stokes waves. Inventiones Mathematicae 166(3), 523–535 (2006)
Constantin, A., Escher, J.: Particle trajectories in solitary water waves. Bull. Am. Math. Soc. 44(3), 423–431 (2007)
Constantin, A., Escher, J.: Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math. 173(1), 559–568 (2011)
Degasperis, A., Procesi, M.: Asymptotic Integrability. Symmetry and Perturbation Theory, pp. 23–37. World Science, Singapore (1999)
Degasperis, A., Holm, D.D., Hone, A.N.W.: A new integrable equation with peakon solutions. Theor. Math. Phys. 133(2), 1463–1474 (2002)
Liu, Y., Yin, Z.: Global existence and blow-up phenomena for the Degasperis–Procesi equation. Commun. Math. Phys. 267(3), 801–820 (2006)
Escher, J., Liu, Y., Yin, Z.: Shock waves and blow-up phenomena for the periodic Degasperis–Procesi equation. Indiana Univ. Math. J. 56, 87–117 (2007)
Liu, Y., Yin, Z.: On the blow-up phenomena for the Degasperis–Procesi equation. Int. Math. Res. Not. 2007(9), 1–22 (2007)
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Yang, S. Wave Breaking Phenomena for the Fornberg–Whitham Equation. J Dyn Diff Equat 33, 1753–1758 (2021). https://doi.org/10.1007/s10884-020-09866-z
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DOI: https://doi.org/10.1007/s10884-020-09866-z