The Local Well-posedness of The Higher-order
Camassa-Holm Equation

DAN-PING DING; XIN LIU

doi:https://doi.org/10.14445/22315373/IJMTT-V4I4P1

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 4 | Issue 4 | Year 2013 | Article Id. IJMTT-V4I4P1 | DOI : https://doi.org/10.14445/22315373/IJMTT-V4I4P1

The Local Well-posedness of The Higher-order Camassa-Holm Equation

DAN-PING DING, XIN LIU

Abstract

In this paper, the local well-posedness of the Cauchy problem for the higher-order Camassa-Holm equation is studied with the initial data in ( ) s H R , s k  by using Bourgain technology.

Keywords

higher-order Camassa-Holm equation, local well-posedness, Fourier transformation.

References

[1] R.Camassa, D.Holm, “An integrable shallow water equation with peaked solitions”, Phys. Rev. Lett. 71 (1993) 1661-1664.
[2] A.Constantin, “On the scattering problem for the Camassa-Holm equation”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 457 (2001) 953-970.
[3] A. Constantin, B. Kolev, “Geodesic flow on the diffeomorphism group of the circle”, Comment. Math. Helv. 78 (4) (2003) 787–804.
[4] G.M. Coclite, H. Holden, and K.H. Karlsen, “Well-posedness of higher-order Camassa–Holm equations”, J. Differential Equations. 246 (2009) 929–963.
[5] J.Bourgain, “Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations I. Schrödinger equations”. Geometric and Functional Analysis. 3 (1993) 107-156.
[6] J.Bourgain, “Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations II. The KdV equation”. Geometric and Functional Analysis. 3 (1993) 209-262.
[7] A.Constantin, J.Escher, “Wave breaking for nonlinear nonlocal shallow water equations”, Acta Math. 181 (1998) 229-243.
[8] E.Wahlén, “On the blow-up of solutions to the periodic Camassa-Holm equation”, Nonlinear differ. equ. appl. 13 (2007) 643–653.
[9] Danping Ding, Peng Lv, “Conservative solutions for higher-order Camassa–Holm equations”, J. Mathematical Physics. 51 (2010) 072701.

Citation :

DAN-PING DING, XIN LIU, "The Local Well-posedness of The Higher-order Camassa-Holm Equation," International Journal of Mathematics Trends and Technology (IJMTT), vol. 4, no. 4, pp. 58-64, 2013. Crossref, https://doi.org/10.14445/22315373/IJMTT-V4I4P1