Global Existence for Two Dimensional Quadratic Derivative Nonlinear Schrödinger Equations

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 λ jk ∈ C. We prove that if the initial data u 0 ∈ H 10 ∩ H 0,10 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 in homogeneous Sobolev space of order one. The proof depends on the energy type estimates, and smoothing property by Doi.

Original languageEnglish
Pages (from-to)732-752
Number of pages21
JournalCommunications in Partial Differential Equations
Volume37
Issue number4
DOIs
Publication statusPublished - 2012 Apr
Externally publishedYes

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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