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Ionescu, Fabio Pusateri","submitted_at":"2012-09-22T00:16:26Z","abstract_excerpt":"We consider the question of global existence of small, smooth, and localized solutions of a certain fractional semilinear cubic NLS in one dimension, $$i\\partial_t u - \\Lambda u = c_0{|u|}^2 u + c_1 u^3 + c_2 u \\bar{u}^2 + c_3 \\bar{u}^3, \\qquad \\Lambda = \\Lambda(\\partial_x) = {|\\partial_x|}^(1/2)$$, where $c_0\\in\\mathbb{R}$ and $c_1,c_2,c_3\\in\\mathbb{C}$. This model is motivated by the two-dimensional water waves equations, which have a somewhat similar structure in the Eulerian formulation, in the case of irrotational flows. 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