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Applicable Analysis
An International Journal
Volume 97, 2018 - Issue 13
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Articles

On the Cauchy problem of generalized Fokas–Olver–Resenau–Qiao equation

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Pages 2246-2268 | Received 18 Mar 2017, Accepted 17 Jul 2017, Published online: 02 Aug 2017

References

  • Camassa R , Holm D . An integrable shallow water equation with peaked solitons. Phys Rev Lett. 1993;71:1661–1664.
  • Fuchessteiner B , Fokas AS . Symplectic structures, their Bäcklund transformations and hereditary symmetries. Phys D. 1981/82;4:47–66.
  • Constantin A , Escher J . Global existence and blow-up for a shallow water equation. Ann Scuola Norm Sup Pisa Cl Sci (4). 1998;26:303–328.
  • Constantin A , Escher J . Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation. Commun Pure Appl Math. 1998;51:475–504.
  • Danchin R . A few remarks on the Camassa--Holm equation. Differ Integr Equ. 2001;14:953–988.
  • Danchin R . A note on well-posedness for Camassa--Holm equation. J Differ Equ. 2003;192:429–444.
  • Rodríguez-Blanco G . On the Cauchy problem for the Camassa--Holm equation. Nonlinear Anal Theory Meth Appl. 2001;46:309–327.
  • Constantin A . Globle existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann Inst Fourier (Grenoble). 2000;50:321–362.
  • Constantin A , Escher J . Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 1998;181:229–243.
  • Constantin A , Molinet L . Global weak solutions for a shallow water equation. Commun Math Phys. 2000;211:45–61.
  • Xin ZP , Zhang P . On the weak solutions to a shallow water equation. Commun Pure Appl Math. 2000;53:1411–1433.
  • Bressan A , Constantin A . Global conservative solutions of the Camassa--Holm equation. Arch Ration Mech Anal. 2007;183:215–239.
  • Bressan A , Constantin A . Global dissipative solutions of the Camassa--Holm equation. Anal Appl (Singap). 2007;5:1–27.
  • Novikov V . Generalizations of the Camassa--Holm equation. J Phys A. 2009;42:342002, p. 14.
  • Himonas A , Holliman C . The Cauchy problem for the Novikov equation. Nonlinearity. 2012;25:449–479.
  • Hone ANW , Wang JP . Integrable peakon equations with cubic nonlinearity. J Phys A. 2008;41:372002, p. 10.
  • Yan W , Li YS , Zhang YM . The Cauchy problem for the Novikov equation. NoDEA Nonlinear Differ Equ Appl. 2013;20:1157–1169.
  • Wu XL , Yin ZY . Well-posedness and global existence for the Novikov equation. Ann Sc Norm Super Pisa Cl Sci (5). 2012;11:707–727.
  • Jiang ZH , Ni LD . Blow-up phenomenon for the integrable Novikov equation. J Math Appl Anal. 2012;385:551–558.
  • Fuchssteiner B . Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation. Phys D. 1996;95:229–243.
  • Olver PJ , Rosenau P . Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys Rev E. 1996;53:1900–1906.
  • Fokas AS . On a class of physically important integrable equations. Phys D. 1995;87:145–150.
  • Qiao ZJ . A new integrable equation with cuspons and W/M-shape-peaks solitons. J Math Phys. 2006;47:112701, p. 9.
  • Qu CZ , Liu XC , Liu Y . Stability of peakons for an integrable modified Camassa--Holm equation with cubic nonlinearity. Commun Math Phys. 2013;322:967–997.
  • Liu XC , Liu Y , Qu CZ . Orbital stability of the train of peakons for an integrable modified Camassa--Holm equation. Adv Math. 2014;255:1–37.
  • Himonas A , Mantzavinos D . The Cauchy problem for the Fokas--Olver--Rosenau--Qiao equation. Nonlinear Anal. 2014;95:499–529.
  • Himonas A , Mantzavinos D . Hölder continuity for the Fokas--Olver--Rosenau--Qiao equation. J Nonlinear Sci. 2014;24:1105–1124.
  • Gui GL , Liu Y , Olver PJ , et al . Wave-breaking and peakons for a modified Camassa--Holm equation. Commun Math Phys. 2013;319:731–759.
  • Fu Y , Gui GL , Liu Y , et al . On the Cauchy problem for the integrable modified Camassa--Holm equation with cubic nonlinearity. J Differ Equ. 2013;255:1905–1938.
  • Wu XL , Guo BL . The exponential decay of solutions and traveling wave solutions for a modified Camassa--Holm equation with cubic nonlinearity. J Math Phys. 2014;55:081504, p. 17.
  • Elena R , Stephen CA . A general family of multi-peakon equations. arXiv:1609.04354v1.
  • Bahouri H , Chemin J , Danchin R . Fourier analysis and nonlinear partial differential equations. Vol. 343, Grundlehren der mathematischen wissenschaften. Heidelberg: Springer; 2011.
  • Chemin J . Perfect incompressible fluids. Vol. 14, Oxford lecture series in mathematics and its applications. New York (NY): The Clarendon Press, Oxford University Press; 1998. p. x+187.
  • Danchin R . Fourier analysis method for PDE’s. Vol. 14, Lecture notes. 2005. Available from: http://perso-math.univ-mlv.fr/users/danchin.raphael/cours/courschine.pdf
  • Chen DF , Li YS , Yan W . On well-posedness of two-component Camassa--Holm system in the critical Besov space. Nonlinear Anal. 2015;120:285–298.
  • Tang H , Liu ZR . The Cauchy problem for a two-component Novikov equation in the critical Besov space. J Math Anal Appl. 2015;423:120–135.
  • Luo W , Yin ZY . Local well-posedness in the critical Besov space and persistence properties for a three-component Camassa--Holm system with N-peakon solutions. Disc Contin Dyn Syst. 2016;36:5047–5066.
  • Yan W , Li YS . The Cauchy problem for the modified two-component Camassa--Holm system in critical Besov space. Ann Inst H Poincaré Anal Non Linéaire. 2015;32:443–469.
  • Yan K , Yin Z . On the Cauchy problem for a two-compnent Degasperis--Procesi system. J Differ Equ. 2012;252:2131–2159.

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