Global existence of small analytic solutions to schrödinger equations with quadratic nonlinearity

Nakao Hayashi, Keiichi Kato

Research output: Contribution to journalArticlepeer-review

13 Citations (Scopus)

Abstract

We study the nonlinear Schrödinger equations {i∂tu + 1/2 Δu = F(u, ∇u, ū, ∇ū), (t, x) ∈ R × Rn u(0, x) = ∈0φ, x ∈ Rn. (*) When n = 3, 4, F : C2n+2 → C is quadratic and ∈0 is sufficiently small, it is shown that small analytic solutions of (*) exist globally in time under the condition |∂uF| + |∂ūF| ≤ C|∇u|. Furthermore we give a global existence theorem when n = 2 and F = λ(∂1u∂2ū - ∂1ū∂2u). We notice that we do not need the condition ∂∂juF is pure imaginary which ensures the use of classical energy estimates.

Original languageEnglish
Pages (from-to)773-798
Number of pages26
JournalCommunications in Partial Differential Equations
Volume22
Issue number5-6
Publication statusPublished - 1997 Dec 1
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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