Motivated by transverse stability issues, we address the time evolution under the KP-II flow of perturbations of a solution which does not decay in all directions, for instance the KdV-line soliton. We study two different types of perturbations: perturbations that are square integrable in and perturbations that are square integrable in . In both cases we prove the global well-posedness of the Cauchy problem associated with such initial data.
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Luc Molinet, Jean-Claude Saut, Nikolay Tzvetkov, Global well-posedness for the KP-II equation on the background of a non-localized solution. Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011), no. 5, pp. 653–676DOI 10.1016/J.ANIHPC.2011.04.004