Nonlinear fractional Schrödinger equations in one dimension

DOI10.1016/J.JFA.2013.08.027zbMATH Open1304.35749arXiv1209.4943OpenAlexW1974658023MaRDI QIDQ477024

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Publication date: 2 December 2014

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Abstract: We consider the question of global existence of small, smooth, and localized solutions of a certain fractional semilinear cubic NLS in one dimension, ${\displaystyle }$ipartial_t u - Lambda u = c_0{|u|}^2 u + c_1 u^3 + c_2 u �ar{u}^2 + c_3 �ar{u}^3, qquad Lambda = Lambda(partial_x) = {|partial_x|}^(1/2)${\displaystyle }$, where $c_{0}inmathbbR$ and $c_1,c_2,c_{3}inmathbbC$. 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. 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.

Full work available at URL: https://arxiv.org/abs/1209.4943

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