Subcritical quadratic nonlinear schrödinger equation

Nakao Hayashi, Pavel I. Naumkin

Research output: Contribution to journalArticlepeer-review


We study the initial value problem for the quadratic nonlinear Schrödinger equation iut+1/2uxx=-4iπtγ-1/2u 2, x R, t > u(1,x) = u 1 (x), x ε R, where γ > 0. Suppose that the Fourier transform û 1 of the initial data u 1 satisfies estimates û1 L∞ ≤ ε, d/d (ei/2ε 2û1(ε))L∞,1 ≤ ε> 0 is sufficiently small. Also suppose that Re(eu/2ε 2 û1(ε))≥Cε5/4 for |ξ| ≤ 1. Assume that γ > 0 is small: γ = O(5/4). Then we prove that there exists a unique solution u ∈ C([1, ∞);L 2) of the Cauchy problem (*). Moreover, the solution u approaches for large time t → +∞ a self-similar solution of the quadratic nonlinear Schrödinger equation (*).

Original languageEnglish
Pages (from-to)969-1007
Number of pages39
JournalCommunications in Contemporary Mathematics
Issue number6
Publication statusPublished - 2011 Dec


  • Asymptotics of solutions
  • quadratic nonlinear Schrödinger equation


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