Nonlinear Fractional Schrödinger Equations in One Dimension
نویسندگان
چکیده
We consider the question of global existence of small, smooth, and localized solutions of a certain fractional semilinear cubic NLS in one dimension, i∂tu − Λu = c0|u| 2 u + c1u 3 + c2uu 2 + c3u 3 , Λ = Λ(∂x) = |∂x| 1 2 , where c0 ∈ R and c1, c2, c3 ∈ C. This model is motivated by the two-dimensional water wave equation, which has a somewhat similar structure in the Eulerian formulation, in the case of irrotational flows. We show that one cannot expect linear scattering, even in this simplified model. More precisely, we identify a suitable nonlinear logarithmic correction, and prove global existence and modified scattering of solutions. CONTENTS
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تاریخ انتشار 2012