×

zbMATH — the first resource for mathematics

On the stable self-similar waves for the Camassa-Holm and Degasperis-Procesi equations. (English) Zbl 1447.35271
Summary: This paper mainly studies the explicit wave-breaking mechanism and dynamical behavior of solutions near the explicit self-similar singularity for the Camassa-Holm and Degasperis-Procesi equations, which can be regarded as a model for shallow water dynamics and arising from the approximation of the Hamiltonian for Euler’s equation in the shallow water regime. We prove that the Camassa-Holm and Degasperis-Procesi equations admit stable explicit self-similar solutions. After that, the nonlinear instability of explicit self-similar solution for the Korteweg-de Vries equation is given.
MSC:
35Q35 PDEs in connection with fluid mechanics
35A21 Singularity in context of PDEs
35B35 Stability in context of PDEs
35C06 Self-similar solutions to PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q53 KdV equations (Korteweg-de Vries equations)
PDF BibTeX XML Cite
Full Text: Euclid
References:
[1] B. Alvarez-Samaniego and D. Lannes,Large time existence for 3D water-waves and asymptotics,Invent. Math., 171 (2009), 165-186. · Zbl 1131.76012
[2] T.B. Benjamin, J.L. Bona, and J.J. Mahony,Model equations for long waves in nonlinear dispersive systems,Philos. Trans. Roy. Soc. London Ser. A., 272 (1972), 47-78. · Zbl 0229.35013
[3] J. Bourgain,Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations II: The KdV equation,Geom. Funct. Anal., 3 (1993), 209-262. · Zbl 0787.35098
[4] L. Brandolese,Local-in-space criteria for blowup in shallow water and dispersive rod equations,Commun. Math. Phys., 330 (2014), 401-414. · Zbl 1294.35089
[5] L. Brandolese and M.F. Cortez,On permanent and breaking waves in hyperelastic rods and rings,J. Funct. Anal., 266 (2014), 6954-6987. · Zbl 1295.35009
[6] A. Bressan and A. Constantin,Global conservative solutions of the Camassa-Holm equation,Arch. Ration. Mech. Anal., 183 (2007), 215-239. · Zbl 1105.76013
[7] A. Bressan and A. Constantin,Global dissipative solutions of the Camassa-Holm equation,Anal. Appl., 5 (2007), 1-27. · Zbl 1139.35378
[8] R. Camassa and D. Holm,An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. · Zbl 0972.35521
[9] X.K. Chang, X.M. Chen, and X.B. Hu,A generalized nonisospectral Camassa-Holm equation and its multipeakon solutions,Adv. Math., 263 (2014), 154-177. · Zbl 1304.35200
[10] G.M. Coclite and K.H. Karlsen,On the well-posedness of the Degasperis-Procesi equation,J. Funct. Anal., 233 (2006), 60-91. · Zbl 1090.35142
[11] A. Constantin,On the Cauchy problem for the periodic Camassa-Holm equation,J. Differential Equations, 141 (1997), 218-235. · Zbl 0889.35022
[12] A. Constantin,The Hamiltonian structure of the Camassa-Holm equation,Exposition. Math., 15 (1997), 53-85.
[13] A. Constantin and J. Escher,Global existence and blow-up for a shallow water equation,Ann. Scuola Norm. Pisa, 26 (1998), 303-328. · Zbl 0918.35005
[14] A. Constantin,Existence of permanent and breaking waves for a shallow water equation: a geometric approach,Ann. Inst. Fourier, 50 (2000), 321-362. · Zbl 0944.35062
[15] A. Constantin,On the scattering problem for the Camassa-Holm equation,Proc. R. Soc. Lond. A., 457 (2001), 953-970. · Zbl 0999.35065
[16] A. Constantin and J. Escher,Wave breaking for nonlinear nonlocal shallow water equations,Acta Math., 181 (1998), 229-243. · Zbl 0923.76025
[17] A. Constantin and L. Molinet,Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61. · Zbl 1002.35101
[18] A. Constantin and A. Strauss,Stability of peakons,Commun. Pure Appl. Math., 53 (2000), 603-610. · Zbl 1049.35149
[19] A. Constantin and W. Strauss,Stability of the Camassa-Holm solitons,. J. Nonlinear Sci., 12 (2002), 415-422. · Zbl 1022.35053
[20] A. Constantin and L. Molinet,Orbital stability of the solitary waves for a shallow water equation,Phys. D., 157 (2001), 75-89. · Zbl 0984.35139
[21] A. Constantin, B. Kolev, and J. Lenells,Integrability of invariant metrics on the Virasoro group,Phys. Lett. A, 350 (2006), 75-80. · Zbl 1195.58006
[22] A. Constantin and W.A. Strauss,Stability of a class of solitary waves in compressible elastic rods,Phys. Lett. A., 270 (2000), 140-148. · Zbl 1115.74339
[23] A. Constantin and D. Lannes,The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,Arch. Ration. Mech. Anal., 192 (2009), 165-186. · Zbl 1169.76010
[24] A. Degasperis and M. Procesi,Asymptotic integrability,In “Symmetry and Perturbation Theory,” edited by A. Degasperis, G. Gaeta, Singapore: World Scientific, 1999, pp. 23-37 · Zbl 0963.35167
[25] A. Degasperis, D.D. Holm, and A.N.W. Hone,A New Integral Equation with Peakon Solutions,Theo. Math. Phys., 133 (2002), 1463-1474.
[26] P.G. Drazin and R.S. Johnson, “Solitons: An Introduction,” Cambridge Univ. Press, Cambridge, 1989. · Zbl 0661.35001
[27] J.Eggers and M.A. Fontelos,The role of self-similarity in singularities of partial differential equations,Nonlinearity, 22 (2009), R1-R44 · Zbl 1152.35300
[28] J. Escher, Y. Liu, and Z.Y. Yin,Global weak solutions and blow-up structure for the Degasperis-Procesi equation,J. Funct. Anal., 241 (2006), 457-485. · Zbl 1126.35053
[29] J. Escher and B. Kolev,The Degasperis-Procesi equation as a non-metric Euler equation,Math. Z., 269 (2011), 1137-1153. · Zbl 1234.35220
[30] S. Hakkaev and K. Kirchev,Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa-Holm equation,Commun. PDE., 30 (2005), 761-781. · Zbl 1076.35098
[31] A. Himonas and C. Holliman,On well-posedness of the Degasperis-Procesi equation, Discrete Contin. Dyn. Syst., 31 (2011), 469-488. · Zbl 1237.35137
[32] A.A. Himonas, C. Holliman, and K. Grayshan,Norm inflation and ill-posedness for the Degasperis-Procesi equation,Commun. PDE., 39 (2014), 2198-2215. · Zbl 1304.35609
[33] A.N.W. Hone and J.P. Wang,Prolongation algebras and Hamiltonian operators for peakon equations,Inverse Problems, 19 (2003), 129-145 · Zbl 1020.35096
[34] R.S. Johnson and Camassa-Holm,Korteweg-de Vries and related models for water waves,J. Fluid Mech., 455 (2002), 63-82. · Zbl 1037.76006
[35] T. Kato,Quasi-linear equations of evolution, with applications to partial differential equations,in “Spectral Theory and Differential Equations (Proc. Sympos.),” Dundee, 1974, in: Lecture Notes in Math., vol. 448, Springer, Berlin, 1975, pp. 25-70, dedicated to Konrad J¬®orgens.
[36] C. Kenig, G. Ponce, and L. Vega,Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle,Comm. Pure Appl. Math., 46 (1993), 527-620. · Zbl 0808.35128
[37] H.Y. Li and W.P. Yan,Asymptotic stability and instability of explicit self-similar waves for a class of nonlinear shallow water equations,Commun. Nonlinear Sci. Numer. Simul., 79 (2019), 104928, 13 pp.
[38] J.B. Li and Y. Zhou,Exact solutions in invariant manifolds of some higher-order models describing nonlinear waves,Qual. Theory Dyn. Syst., 18 (2019), 183-199. · Zbl 1439.34004
[39] J.B. Li and G.R. Chen,More on bifurcations and dynamics of traveling wave solutions for a higher-order shallow water wave equation,Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1950014, 13 pp. · Zbl 1415.34074
[40] Y. Liu and Z. Yin,Global existence and blow-up phenomena for the Degasperis-Procesi equation,Comm. Math. Phys., 267 (2006), 801-820. · Zbl 1131.35074
[41] H. Lundmark,Formation and dynamics of shock waves in the Degasperis-Procesi equation,J. Nonlinear Sci., 17 (2007), 169-198. · Zbl 1185.35194
[42] H. Mckean,Breakdown of the Camassa-Holm equation,Commun. Pure Appl. Math., 57 (2004), 416-418. · Zbl 1052.35130
[43] A. Pazy, “Semigroups of Linear Operators and Applications to Partial Differential Equations,” Springer-Verlag, New York, 1983. · Zbl 0516.47023
[44] T. Tao,Low-regularity global solutions to nonlinear dispersive equations,in “Surveys in Analysis and Operator Theory,” Canberra, 2001, in: Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 40, Austral. Nat. Univ., Canberra, 2002, pp. 19-48.
[45] G.B. Whitham, “Linear and Nonlinear Waves,” Wiley, New York · Zbl 0373.76001
[46] Z. Yin,On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (2003), 649-666. · Zbl 1061.35142
[47] Z.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.