Quadratic derivative nonlinear Schrödinger equations in two space dimensions

Fernando Bernal-Vílchis, Nakao Hayashi, Pavel I. Naumkin

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

Abstract

We study the global in time existence of small classical solutions to the nonlinear Schrödinger equation with quadratic interactions of derivative type in two space dimensions, where the quadratic nonlinearity has the form N(∇u,∇v) = ∑k,l=1,2λkl (∂ku)(∂lv) with λ ε C. We prove that if the initial data u0 ε H6 ∩ H3,3 satisfy smallness conditions in the weighted Sobolev norm, then the solution of the Cauchy problem (0.1) exists globally in time. Furthermore we prove the existence of the usual scattering states. The proof depends on the energy type estimates, smoothing property by Doi, and method of normal forms by Shatah.

Original languageEnglish
Pages (from-to)329-355
Number of pages27
JournalNonlinear Differential Equations and Applications
Volume18
Issue number3
DOIs
Publication statusPublished - 2011 Jun

Keywords

  • Global existence
  • Nonlinear Schrödinger equations
  • Quadratic nonlinearities
  • Two spatial dimensions

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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