Abstract
We prove that the flow map of the Kadomtsev–Petviashvili-I (KP-I) equation is not uniformly continuous on bounded sets of the natural energy space.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Burq N., Gérard P. and Tzvetkov N. (2003). Two singular dynamics of the non-linear Schrödinger equation on a plane domain. Geom. Funct. Anal. 13(1): 1–19
Burq N., Gérard P. and Tzvetkov N. (2002). An instability property of the non-linear Schrödinger equation on S d. Math. Res. Lett. 9(2–3): 323–335
Christ M., Colliander J. and Tao T. (2003). Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations. Am. J. Math. 125(6): 1235–1293
Iório, R.J. Jr., Nunes, W.V.L.: On equations of KP-type. Proc. R. Soc. Edinburgh Sect. A 725–743 (1998)
Ionescu, A., Kenig, C.: Global well-posedness of the Benjamin–Ono equation in low-regularity spaces. arXiv:math.AP/0508632 (2005)
Kenig C. (2004). On the local and global well-posedness for the KP-I equation. Ann. IHP Anal. Non linéaire 21: 827–838
Koch H. and Tzvetkov N. (2005). Non-linear wave interactions for the Benjamin–Ono equation. IMRN 30: 1833–1847
Molinet, L., Saut, J.C., Tzvetkov, N.: Global well-posedness for the KP-I equation. Math. Ann. 324, 255–275 (2002). Correction: Math. Ann. 328, 707–710 (2004)
Molinet L., Saut J.C. and Tzvetkov N. (2002). Well-posedness and ill-posedness for the Kadomtsev– Petviashvili-I equation. Duke Math. J. 115: 353–384
Tao T. (2004). Global well-posedness of the Benjamin–Ono equation in H 1(R). J. Hyperbolic Differ. Equ. 1: 27–49
Tom M.M. (1996). On a generalized Kadomtsev–Petviashvili equation. Contemp. Math. AMS 200: 193–210
Tzvetkov, N.: Ill-posedness issues for non-linear dispersive equations. GAKUTO international series mathematical sciences and applications, vol. 27, September, 2006. aiXiv:math.AP/0411455
Zakharov V. and Schulman E. (1980). Degenerative dispersion laws, motion invariants and kinetic equations. Physica D1: 192–202
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Koch, H., Tzvetkov, N. On finite energy solutions of the KP-I equation. Math. Z. 258, 55–68 (2008). https://doi.org/10.1007/s00209-007-0156-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-007-0156-x