Publication Cover
Applicable Analysis
An International Journal
Volume 100, 2021 - Issue 6
184
Views
0
CrossRef citations to date
0
Altmetric
Articles

Blow-up scenario for a generalized Camassa–Holm equation with both quadratic and cubic nonlinearity

Pages 1180-1197 | Received 28 Feb 2019, Accepted 21 Jun 2019, Published online: 04 Jul 2019

References

  • Constantin A, Lannes D. The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations. Arch Ration Mech Anal. 2009;192(1):165–186. doi: 10.1007/s00205-008-0128-2
  • Ionescu-Kruse D. Variational derivation of the Camassa–Holm shallow water equation. J Nonlinear Math Phys. 2007;14(3):303–312. doi: 10.2991/jnmp.2007.14.3.1
  • Johnson RS. Camassa–Holm, Korteweg–de Vries and related models for water waves. J Fluid Mech. 2002;455:63–82. doi: 10.1017/S0022112001007224
  • Xia B, Qiao Z, Li J. An integrable system with peakon, complex peakon, weak kink, and kink-peakon interactional solutions. Commun Nonlinear Sci Numer Simul. 2018;63:292–306. doi: 10.1016/j.cnsns.2018.03.019
  • DeSanto JA, editor. Mathematical and numerical aspects of wave propagation. Philadelphia (PA): Society for Industrial and Applied Mathematics (SIAM); 1998.
  • Olver PJ, Rosenau P. Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys Rev E (3). 1996;53(2):1900–1906. doi: 10.1103/PhysRevE.53.1900
  • Fokas AS. On a class of physically important integrable equations. Physica D. 1995;87(1–4):145–150. The nonlinear Schrödinger equation (Chernogolovka, 1994). doi: 10.1016/0167-2789(95)00133-O
  • Fuchssteiner B. Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa–Holm equation. Physica D. 1996;95(3–4):229–243. doi: 10.1016/0167-2789(96)00048-6
  • Fuchssteiner B, Fokas AS. Symplectic structures, their Bäcklund transformations and hereditary symmetries. Physica D. 1981/82;4(1):47–66. doi: 10.1016/0167-2789(81)90004-X
  • Olver PJ, Rosenau P. Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys Rev E (3). 1996;53(2):1900–1906. doi: 10.1103/PhysRevE.53.1900
  • Qiao Z. A new integrable equation with cuspons and W/M-shape-peaks solitons. J Math Phys. 2006;47(11):112701. doi: 10.1063/1.2365758
  • Gui G, Liu Y, Olver PJ, et al. Wave-breaking and peakons for a modified Camassa–Holm equation. Commun Math Phys. 2013;319(3):731–759. doi: 10.1007/s00220-012-1566-0
  • Fu Y, Gui G, Liu Y, et al. On the Cauchy problem for the integrable modified Camassa–Holm equation with cubic nonlinearity. J Differ Equ. 2013;255(7):1905–1938. doi: 10.1016/j.jde.2013.05.024
  • Danchin R. A few remarks on the Camassa–Holm equation. Differ Int Equ. 2001;14(8):953–988.
  • Himonas AA, Mantzavinos D. Hölder Continuity for the Fokas-Olver-Rosenau-Qiao Equation. J Nonlinear Sci. 2014;24(6):1105–1124. doi: 10.1007/s00332-014-9212-y
  • Camassa R, Holm DD. An integrable shallow water equation with peaked solitons. Phys Rev Lett. 1993;71(11):1661–1664. doi: 10.1103/PhysRevLett.71.1661
  • Dai H-H. Model equations for nonlinear dispersive waves in a compressible Mooney–Rivlin rod. Acta Mech. 1998;127(1–4):193–207. doi: 10.1007/BF01170373
  • Constantin A. On the scattering problem for the Camassa–Holm equation. R Soc Lond Proc Ser A Math Phys Eng Sci. 2001;457(2008):953–970. doi: 10.1098/rspa.2000.0701
  • Constantin A. The trajectories of particles in Stokes waves. Invent Math. 2006;166(3):523–535. doi: 10.1007/s00222-006-0002-5
  • Constantin A, Escher J. Particle trajectories in solitary water waves. Bull Amer Math Soc (N.S). 2007;44(3):423–431. doi: 10.1090/S0273-0979-07-01159-7
  • Constantin A, Escher J. Analyticity of periodic traveling free surface water waves with vorticity. Ann Math (2). 2011;173(1):559–568. doi: 10.4007/annals.2011.173.1.12
  • Constantin A, Escher J. Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 1998;181(2):229–243. doi: 10.1007/BF02392586
  • Constantin A, Strauss WA. Stability of the Camassa–Holm solitons. J Nonlinear Sci. 2002;12(4):415–422. doi: 10.1007/s00332-002-0517-x
  • Constantin A, Strauss WA. Stability of peakons. Commun Pure Appl Math. 2000;53(5):603–610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L
  • Bressan A, Constantin A. Global conservative solutions of the Camassa–Holm equation. Arch Ration Mech Anal. 2007;183(2):215–239. doi: 10.1007/s00205-006-0010-z
  • Bressan A, Constantin A. Global dissipative solutions of the Camassa–Holm equation. Anal Appl (Singapore). 2007;5(1):1–27. doi: 10.1142/S0219530507000857
  • Hakkaev S, Kirchev K. Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa–Holm equation. Commun Partial Differ Equ. 2005;30(4–6):761–781. doi: 10.1081/PDE-200059284
  • Alexandrou Himonas A, Holliman C. The Cauchy problem for a generalized Camassa–Holm equation. Adv Differ Equ. 2014;19(12):161–200.
  • Lai S, Wu Y. The local well-posedness and existence of weak solutions for a generalized Camassa–Holm equation. J Differ Equ. 2010;248(8):2038–2063. doi: 10.1016/j.jde.2010.01.008
  • Li M, Yin Z. Blow-up phenomena and local well-posedness for a generalized Camassa–Holm equation with cubic nonlinearity. Nonlinear Anal. 2017;151:208–226. doi: 10.1016/j.na.2016.12.003
  • Mustafa OG. On the Cauchy problem for a generalized Camassa–Holm equation. Nonlinear Anal. 2006;64(6):1382–1399. doi: 10.1016/j.na.2005.06.042
  • Novikov V. Generalizations of the Camassa–Holm equation. J Phys A. 2009;42(34):342002. doi: 10.1088/1751-8113/42/34/342002
  • Yin Z. On the blow-up scenario for the generalized Camassa–Holm equation. Commun Partial Differ Equ. 2004;29(56):867–877. doi: 10.1081/PDE-120037334
  • Liu X, Qiao Z, Yin Z. On the Cauchy problem for a generalized Camassa–Holm equation with both quadratic and cubic nonlinearity. Commun Pure Appl Anal. 2014;13(3):1283–1304. doi: 10.3934/cpaa.2014.13.1283
  • Bahouri H, Chemin J-Y, Danchin R. Fourier analysis and nonlinear partial differential equations. Heidelberg: Springer; 2011. (Grundlehren der Mathematischen Wissenschaften [Fundamental principles of mathematical sciences]; vol. 343).
  • Gui G, Liu Y. On the global existence and wave-breaking criteria for the two-component Camassa–Holm system. J Funct Anal. 2010;258(12):4251–4278. doi: 10.1016/j.jfa.2010.02.008
  • Constantin A. Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann Inst Fourier (Grenoble). 2000;50(2):321–362. doi: 10.5802/aif.1757
  • Constantin A, Kappeler T, Kolev B, et al. On geodesic exponential maps of the Virasoro group. Ann Global Anal Geom. 2007;31(2):155–180. doi: 10.1007/s10455-006-9042-8
  • Constantin A, Kolev B. Geodesic flow on the diffeomorphism group of the circle. Comment Math Helv. 2003;78(4):787–804. doi: 10.1007/s00014-003-0785-6
  • Kolev B. Lie groups and mechanics: an introduction. J Nonlinear Math Phys. 2004;11(4):480–498. doi: 10.2991/jnmp.2004.11.4.5

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.