Abstract
This paper mainly investigate the Cauchy problem for the generalised two-component Camassa–Holm type system, which includes the celebrated Camassa–Holm equation, Degasperis equation, Novikov equation, and the two-component cross-coupled Camassa–Holm system, Novikov system as special cases. Firstly, the local well-posedness of the system in nonhomogeneous Besov spaces \(B^{s}_{l,r}(\mathbb {R})\times B^{s}_{l,r}(\mathbb {R})\) with \(l,r\in [1,\infty ]\), \(s>\max \{2+1/l,5/2\}\) is established by using the Littlewood–Paley theory and transport equations theory. Moreover, we verify the blow-up occurs for this system only in the form of breaking waves. Finally, the waltzing peakons for the system and some numerical experiments to illustrate our results are performed.
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Bahouri, H., Chemin, J.Y., Danchin, R.: Fourier analysis and nonlinear partial differential equations. In: Grundlehren der Mathematischen Wissenschaften, 343. Springer, Heidelberg (2011)
Costantin, A., Lannes, D.: The hydrodynamical of relevance of Camassa–Holm and Degasperis–Procesi equations. Arch. Ration. Mech. Anal. 192, 165–186 (2009)
Cotter, C.J., Holm, D.D., Ivanov, R.I., Percival, J.R.: Waltzing peakons and compacton pairs in a cross-coupled Camassa–Holm equation. J. Phys. A Math. Theor. 44, 2065–2088 (2016)
Danchin, R.: Fourier analysis methods for PDEs. Lect. Notes 14, 1–91 (2005)
Danchinz, R.: A few remarks on the Camassa-equation. Differ. Integr. Equ. 14, 953–988 (2001)
Degasperis, A., Procesi, M.: Asymptotic integrability. In: Symmertry and Perturbation Theroy, pp. 23–37. Word Scientific, Singapore (1999)
Degasperis, A., Holm, D.D., Hone, A.N.W.: A new integral equation with peakon solutions. Theor. Math. Phys. 133, 1463–1474 (2002)
Feng, Q., Meng, F., Zheng, B.: Gronwall–Bellman type nonlinear delay integral inequalities on times scales. J. Math. Anal. Appl. 382, 772–784 (2011)
Geng, X., Xue, B.: An extension of integrable peakon equations with cubic nonlinearity. Nonlinearity 22, 1847–1856 (2009)
Gui, G., Liu, Y.: On the global existence and wave-breaking criteria for the two-component Camassa–Holm equation. J. Funct. Anal. 258, 4251–4278 (2010)
Hao, X., Liu, L., Wu, Y., Sun, Q.: Positive solutions for nonlinear \(n\)th-order singular eigenvalue problem with nonlocal conditions. Nonlinear Anal. 73, 1653–1662 (2010)
He, X., Qian, A., Zou, W.: Existence and concentration of positive solutions for quasi-linear Schrodinger equations with critical growth. Nonlinearity 26, 3137–3168 (2013)
Henry, A., Holm, D.D., Ivanov, R.I.: On the persistence properties of the cross-coupled Camassa–Holm system. J. Geom. Symmetry Phys. 32, 1–13 (2013)
Himonas, A.A., Mantzavinos, D.: The initial value problem for a Novikov system. J. Math. Phys. 57, 071503 (2016)
Ivanov, R.I.: Water waves and integrability. Philos. Trans. R. Soc. Lond. A 3(65), 1498–1521 (2007)
Li, F.: Limit behavior of the solution to nonlinear viscoelastic Marguerre–von Karman shallow shell system. J. Differ. Equ. 249, 1241–1257 (2010)
Lundmark, H., Szmigielski, J.: An inverse spectral problem related to the Geng–Xue two-component peakon equation. ar**v:1304.0854v1
Ma, X., Wang, P., Wei, W.: Constant mean curvature surfaces and mean curvature flow with non-zero Neumann boundary conditions on strictly convex domains. J. Funct. Anal. 274, 252–277 (2018)
Mi, Y., Mu, C., Tao, W.: On the Cauchy problem for the two-component Novikov equation. Adv. Math. Phys. 2, 105–121 (2013)
Tang, H., Liu, Z.: The Cauchy problem for a two-componment Novikov equation in the critical Besov space. J. Math. Anal. Appl. 423, 120–135 (2015)
Taylor, M.: Commutator estimates. Proc. Am. Math. Soc. 131, 1501–1507 (2003)
Triebel, H.: Theory of function spaces. Monogr. Math. 265, 249–263 (1983)
Wang, P.: The concavity of the Gaussian curvature of the convex level sets of minimal surfaces with respect to the height. Pacific J. Math. 267, 489–509 (2014)
Wang, P., Zhang, D.: Convexity of level sets of minimal graph on space form with nonnegative curvature. J. Differ. Equ. 262, 5534–5564 (2017)
Wang, P., Zhao, L.: Some geometrical properties of convex level sets of minimal graph on 2-dimensional Riemannian manifolds. Nonlinear Anal. 130, 1–17 (2016)
Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)
Yan, W., Li, Y., Zhang, Y.: The Cauchy problem for the generalized Camassa–Holm equation in Besov space. J. Differ. Equ. 256, 2876–2901 (2014)
Zhou, S.: Well-posedness and blow up phenomena for a cross-coupled Camassa–Holm equation with waltzing peakons and compacton pairs. J. Evol. Equ. 14, 727–747 (2014)
Zhou, S., Mu, C.: The properties of solutions for a generalized b-family equation with higher-order nonlinearities and peakons. J. Nonlinear Sci. 23, 863–889 (2013)
Zhou, S., Qiao, Z., Mu, C., Wei, L.: Continuity and asymptotic behaviors for a shallow water wave model with moderate amplitude. J. Differ. Equ. 263, 910–933 (2017)
Acknowledgements
The authors are very grateful to the anonymous reviewers and editors for their careful reading and useful suggestions, which greatly improved the presentation of the paper. This work is partly supported by the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJ1703043), Natural Science Foundation of Chongqing (Grant No. cstc2017jcyjAX0123) and Scientific Research Innovation Project for Graduate Student of Chongqing Normal University (Grant No. YKC18038).
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Yang, L., Zhang, B. & Zhou, S. A new generalised two-component Camassa–Holm type system with waltzing peakons and wave breaking. Nonlinear Differ. Equ. Appl. 25, 37 (2018). https://doi.org/10.1007/s00030-018-0528-4
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DOI: https://doi.org/10.1007/s00030-018-0528-4