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Local-in-space blow-up and symmetry of traveling wave solutions to a generalized two-component Dullin–Gottwald–Holm system

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Abstract

In this paper, we are concerned with the Cauchy problem for a generalized two-component Dullin–Gottwald–Holm system arising from the shallow water regime with nonzero constant vorticity. We provide new sufficient conditions on the initial data which lead to the local-in-space blow-up. In addition, it is shown that horizontally symmetric weak solutions to this system must be traveling wave solutions.

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References

  1. Brandolese, L.: Local-in-space criteria for blowup in shallow water and dispersive rod equations. Commun. Math. Phys. 330, 401–414 (2014)

    MathSciNet  MATH  Google Scholar 

  2. Brandolese, L., Cortez, M.F.: Blowup issues for a class of nonlinear dispersive wave equations. J. Differ. Equ. 256, 3981–3998 (2014)

    MathSciNet  MATH  Google Scholar 

  3. Brandolese, L., Cortez, M.F.: On permanent and breaking waves in hyperelastic rods and rings. J. Funct. Anal. 266, 6954–6987 (2014)

    MathSciNet  MATH  Google Scholar 

  4. Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)

    MathSciNet  MATH  Google Scholar 

  5. Cai, H., Chen, G., Shen, Y.N.: Lipschitz metric for conservative solutions of the two-component Camassa–Holm system. Z. Angew. Math. Phys. 68, 5 (2017)

    MathSciNet  MATH  Google Scholar 

  6. Chen, C.X., Yan, Y.H.: On the wave breaking phenomena for the generalized periodic two-component Dullin–Gottwald–Holm system. J. Math. Phys. 53, 103709 (2012)

    MathSciNet  MATH  Google Scholar 

  7. Chen, R.M., Fan, L.L., Gao, H.J., Liu, Y.: Breaking waves and solitary waves to the rotation-two-component Camassa–Holm system. SIAM J. Math. Anal. 49, 3573–3602 (2017)

    MathSciNet  MATH  Google Scholar 

  8. Chen, Y., Gao, H.J., Liu, Y.: On the Cauchy problem for the two-component Dullin–Gottwald–Holm system. Discrete Contin. Dyn. Syst. 33, 3407–3441 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Constantin, A.: Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier (Grenoble) 50, 321–362 (2000)

    MathSciNet  MATH  Google Scholar 

  10. Constantin, A., Escher, J.: On the blow-up rate and the blow-up set of breaking waves for a shallow water equation. Math. Z. 233, 75–91 (2000)

    MathSciNet  MATH  Google Scholar 

  11. Constantin, A., Ivanov, R.: On an integrable two-component Camassa-Holm shallow water system. Phys. Lett. A 372, 7129–7132 (2008)

    MathSciNet  MATH  Google Scholar 

  12. Constantin, A., Strauss, W.A.: Stability of peakons. Commun. Pure Appl. Math. 53, 603–610 (2000)

    MathSciNet  MATH  Google Scholar 

  13. Dullin, H.R., Gottwald, G.A., Holm, D.D.: An integrable shallow water equation with linear and nonlinear dispersion. Phys. Rev. Lett. 87, 194501 (2001)

    Google Scholar 

  14. Ehrnström, M., Holden, H., Raynaud, X.: Symmetric waves are traveling waves. Int. Math. Res. Not. 2009, 4578–4596 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Escher, J., Lechtenfeld, O., Yin, Z.Y.: Well-posedness and blow-up phenomena for the 2-component Camassa–Holm equation. Discrete Contin. Dyn. Syst. 19, 493–513 (2007)

    MathSciNet  MATH  Google Scholar 

  16. Fuchssteiner, B., Fokas, A.S.: Symplectic structures, their Bäcklund transformations and hereditary symmetries. Phys. D 4, 47–66 (1981)

    MathSciNet  MATH  Google Scholar 

  17. Grunert, K.: Blow-up for the two-component Camassa–Holm system. Discrete Contin. Dyn. Syst. 35, 2041–2051 (2015)

    MathSciNet  MATH  Google Scholar 

  18. Grunert, K., Holden, H., Raynaud, X.: Global solutions for the two-component Camassa–Holm system. Commun. Partial Differ. Equ. 37, 2245–2271 (2012)

    MathSciNet  MATH  Google Scholar 

  19. Guan, C.X., Yin, Z.Y.: Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system. J. Differ. Equ. 248, 2003–2014 (2010)

    MathSciNet  MATH  Google Scholar 

  20. Guan, C.X., Yin, Z.Y.: Global weak solutions for a two-component Camassa–Holm shallow water system. J. Funct. Anal. 260, 1132–1154 (2011)

    MathSciNet  MATH  Google Scholar 

  21. Guan, C.X., Zhu, H.: Well-posedness of a class of solutions to an integrable two-component Camassa-Holm system. J. Math. Anal. Appl. 470, 413–433 (2019)

    MathSciNet  MATH  Google Scholar 

  22. Gui, G.L., Liu, Y.: On the global existence and wave-breaking criteria for the two-component Camassa–Holm system. J. Funct. Anal. 258, 4251–4278 (2010)

    MathSciNet  MATH  Google Scholar 

  23. Gui, G.L., Liu, Y.: On the Cauchy problem for the two-component Camassa–Holm system. Math. Z. 268, 45–66 (2011)

    MathSciNet  MATH  Google Scholar 

  24. Guo, F., Gao, H.J., Liu, Y.: On the wave-breaking phenomena for the two-component Dullin–Gottwald–Holm system. J. Lond. Math. Soc. 86, 810–834 (2012)

    MathSciNet  MATH  Google Scholar 

  25. Guo, Z.G., Ni, L.D.: Wave breaking for the periodic weakly dissipative Dullin–Gottwald–Holm equation. Nonlinear Anal. 74, 965–973 (2011)

    MathSciNet  MATH  Google Scholar 

  26. Hakkaev, S.: Stability of peakons for an integrable shallow water equation. Phys. Lett. A 354, 137–144 (2006)

    Google Scholar 

  27. Han, Y.W., Guo, F., Gao, H.J.: On solitary waves and wave-breaking phenomena for a generalized two-component integrable Dullin–Gottwald–Holm system. J. Nonlinear Sci. 23, 617–656 (2013)

    MathSciNet  MATH  Google Scholar 

  28. He, F.Y., Wei, Z.Q.: A remark on wave breaking for the Dullin–Gottwald–Holm equation. Appl. Anal. 96, 1928–1934 (2017)

    MathSciNet  MATH  Google Scholar 

  29. Hoang, D.T.: The local criteria for blowup of the Dullin–Gottwald–Holm equation and the two-component Dullin–Gottwald–Holm system. Ann. Fac. Sci. Toulouse Math. 25, 995–1012 (2016)

    MathSciNet  MATH  Google Scholar 

  30. Ivanov, R.: Two-component integrable systems modelling shallow water waves: the constant vorticity case. Wave Mot. 46, 389–396 (2009)

    MathSciNet  MATH  Google Scholar 

  31. Li, M.J., Zhang, Q.T.: Uniqueness of conservative solutions to the two-component Camassa–Holm system via characteristics. Commun. Math. Sci. 14, 1581–1597 (2016)

    MathSciNet  MATH  Google Scholar 

  32. Liu, X.X., Yin, Z.Y.: Orbital stability of the sum of \(N\) peakons for the Dullin–Gottwald–Holm equation. Nonlinear Anal. Real World Appl. 13, 2414–2422 (2012)

    MathSciNet  MATH  Google Scholar 

  33. Liu, X.X., Yin, Z.Y.: Local well-posedness and stability of solitary waves for the two-component Dullin–Gottwald–Holm system. Nonlinear Anal. 88, 1–15 (2013)

    MathSciNet  MATH  Google Scholar 

  34. Liu, Y.: Global existence and blow-up solutions for a nonlinear shallow water equation. Math. Ann. 335, 717–735 (2006)

    MathSciNet  MATH  Google Scholar 

  35. Novruzov, E.: Blow-up phenomena for the weakly dissipative Dullin–Gottwald–Holm equation. J. Math. Phys. 54, 092703 (2013)

    MathSciNet  MATH  Google Scholar 

  36. Olver, P.J., Rosenau, P.: Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys. Rev. E 53, 1900–1906 (1996)

    MathSciNet  Google Scholar 

  37. Tian, L.X., Gui, G.L., Liu, Y.: On the well-posedness problem and the scattering problem for the Dullin–Gottwald–Holm equation. Commun. Math. Phys. 257, 667–701 (2005)

    MathSciNet  MATH  Google Scholar 

  38. Wang, Y., Zhu, M.: On the persistence and blow up for the generalized two-component Dullin–Gottwald–Holm system. Monatsh. Math. 191, 377–394 (2020)

    MathSciNet  MATH  Google Scholar 

  39. Wei, L., Wang, Y.: Symmetry analysis, conserved quantities and applications to a dissipative DGH equation. J. Differ. Equ. 266, 3189–3208 (2019)

    MathSciNet  MATH  Google Scholar 

  40. Wei, L., Wang, Y., Zhang, H.Y.: Breaking waves and persistence property for a two-component Camassa–Holm system. J. Math. Anal. Appl. 445, 1084–1096 (2017)

    MathSciNet  MATH  Google Scholar 

  41. Yan, K., Yin, Z.Y.: On the solutions of the Dullin–Gottwald–Holm equation in Besov spaces. Nonlinear Anal. Real World Appl. 13, 2580–2592 (2012)

    MathSciNet  MATH  Google Scholar 

  42. Yang, S.J., Xu, T.Z.: Local-in-space blow-up and symmetric waves for a generalized two-component Camassa–Holm system. Appl. Math. Comput. 347, 514–521 (2019)

    MathSciNet  MATH  Google Scholar 

  43. Yin, Z.Y.: Well-poseness, blowup, and global existence for an integrable shallow water equation. Discrete Contin. Dyn. Syst. 11, 393–411 (2004)

    MathSciNet  Google Scholar 

  44. Zhang, P.Z., Liu, Y.: Stability of solitary waves and wave-breaking phenomena for the two-component Camassa–Holm system. Int. Math. Res. Not. 2010, 1981–2021 (2010)

    MathSciNet  MATH  Google Scholar 

  45. Zhang, S.H., Yin, Z.Y.: On the blow-up phenomena of the periodic Dullin–Gottwald–Holm equation. J. Math. Phys. 49, 113504 (2008)

    MathSciNet  MATH  Google Scholar 

  46. Zhang, S.H., Yin, Z.Y.: Global weak solutions for the Dullin–Gottwald–Holm equation. Nonlinear Anal. 72, 1690–1700 (2010)

    MathSciNet  MATH  Google Scholar 

  47. Zhang, Z.Y., Huang, J.H., Sun, M.B.: Blow-up phenomena for the weakly dissipative Dullin–Gottwald–Holm equation revisited. J. Math. Phys. 56, 092701 (2015)

    MathSciNet  MATH  Google Scholar 

  48. Zhou, Y.: Blow-up of solutions to the DGH equation. J. Funct. Anal. 250, 227–248 (2007)

    MathSciNet  MATH  Google Scholar 

  49. Zhu, M., Xu, J.X.: On the wave-breaking phenomena for the periodic two-component Dullin–Gottwald–Holm system. J. Math. Anal. Appl. 391, 415–428 (2012)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is supported by the Scientific Research Foundation of Educational Committee of Yunnan Province (No. 2019J0735), and the Joint Special Funds of the Basic Research Foundation of Yunnan Province (No. 2019FH001-080).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

    Wenguang Cheng & Tianzhou Xu

  2. Department of Mathematics, Yuxi Normal University, Yuxi, 653100, People’s Republic of China

    Wenguang Cheng

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  1. Wenguang Cheng
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Correspondence to Wenguang Cheng.

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Communicated by Adrian Constantin.

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Cheng, W., Xu, T. Local-in-space blow-up and symmetry of traveling wave solutions to a generalized two-component Dullin–Gottwald–Holm system. Monatsh Math 193, 573–589 (2020). https://doi.org/10.1007/s00605-020-01411-w

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