×

Well-posedness for the ZK equation in a cylinder and on the background of a KdV soliton. (English) Zbl 1198.35228

Summary: We establish the well-posedness of two Cauchy problems for the two-dimensional Zakharov-Kuznetsov equation motivated by the study of transverse stability properties of the \(N\)-soliton \(\phi ^N\) of the Korteweg-de Vries equation. They differ by the functional setting: Sobolev spaces \(H^s(\mathbb R \times \mathbb T), s > \frac32\) for the first one, \(\phi ^N + H^{1}(\mathbb R^{2})\) for the second one. In the latter case, the solution is shown to be global.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B10 Periodic solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Biagioni H., Progr. Nonlinear Diff. Eqs. Appl. 54 pp 181– (2001)
[2] Drazin P.G., Solitons: An Introduction (1989)
[3] Ionescu A.D., Ann. of Math. Stud. 163 pp 181– (2007)
[4] DOI: 10.1007/s00222-008-0115-0 · Zbl 1188.35163
[5] Faminskii A.V., Diff. Eq. 31 pp 1002– (1995)
[6] DOI: 10.1002/cpa.3160410704 · Zbl 0671.35066
[7] DOI: 10.1016/j.anihpc.2003.12.002 · Zbl 1072.35162
[8] DOI: 10.1002/cpa.3160460405 · Zbl 0808.35128
[9] DOI: 10.1017/S0022377800000428
[10] DOI: 10.1137/080739173 · Zbl 1197.35242
[11] DOI: 10.3934/dcds.2009.24.547 · Zbl 1170.35086
[12] DOI: 10.1137/1018076 · Zbl 0333.35021
[13] DOI: 10.1007/s00220-007-0243-1 · Zbl 1160.35065
[14] DOI: 10.1016/j.matpur.2008.07.004 · Zbl 1159.35063
[15] DOI: 10.1016/j.anihpc.2007.09.006 · Zbl 1169.35374
[16] Stein E.M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals (1993) · Zbl 0821.42001
[17] Stein E.M., Introduction to Fourier Analysis on Euclidean Spaces (1971) · Zbl 0232.42007
[18] Zakharov V.E., Sov. Phys. JETP. 39 pp 285– (1974)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.