Nonlinear fractional Schrödinger equations in one dimension

Alexandru D. Ionescu, Fabio Giuseppe Pusateri

Research output: Contribution to journalArticle

81 Scopus citations

Abstract

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+c3-3,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.

Original languageEnglish (US)
Pages (from-to)139-176
Number of pages38
JournalJournal of Functional Analysis
Volume266
Issue number1
DOIs
StatePublished - Jan 1 2014

All Science Journal Classification (ASJC) codes

  • Analysis

Keywords

  • Global regularity
  • Modified scattering

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