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.


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
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[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)
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