## 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

### Keywords:

KdV N-solitons; Strichartz estimates; well-posedness
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### References:

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