Nonlinear fractional Schrödinger equations in one dimension

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|2u+c1u3+c2uu¯2+c3u¯3,Λ=Λ(∂x)=|∂x|12, where c0∈R and c1,c2,c3∈C. This model is motivated by the two-dimensional water wave equation, which...

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Bibliographic Details
Published in:Journal of functional analysis 2014-01, Vol.266 (1), p.139-176
Main Authors: Ionescu, Alexandru D., Pusateri, Fabio
Format: Article
Language:English
Subjects:
Global regularity
Modified scattering
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Summary: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|2u+c1u3+c2uu¯2+c3u¯3,Λ=Λ(∂x)=|∂x|12, 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.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2013.08.027