Global existence of small solutions to the generalized derivative nonlinear Schrödinger equation

Nakao Hayashi, Changxing Miao, Pavel I. Naumkin

Research output: Contribution to journalArticlepeer-review

11 Citations (Scopus)

Abstract

We study the global in time existence of small solutions to the generalized derivative nonlinear Schrödinger equations of the form i∂tu + (1/2)Δu = N(u, ∇u, ū ∇ū), (t, x) ∈ R × Rn, (A) u(0, x) = u0(x), x ∈ Rn, where the space dimension n ≥ 3, the initial data u0 are sufficiently small, ū is the complex conjugate of u and the nonlinear term N is a smooth complex valued function C × Cn × C × Cn → C. We assume that N is a quadratic function in the neighborhood of the origin and always includes at least one derivative, that is, |N(u, w, ū w̄)| ≤ C|w|(|u| + |w|), for small u and w in the case of space dimensions n = 3, 4. As a typical example we consider the case of the polynomial type nonlinearity of the form N(u, w, ū, w̄) = ∑ λαβγuα1ū α2wβγ 2 ≤ |α| + |β| + |γ| ≤ l m ≤ |β| + |γ| ≤ l with w = (wj)1≤j≤n, λαβγ ∈ C, l ≥ 2, m ≥ 1 for n = 3, 4, and m ≥ 0 for n ≥ 5. We prove the global existence of solutions to the Cauchy problem (A) under the condition that the initial data u0 ∈ H[n/2]+5,0 ∩H[/2]+3,2, where Hm,s = {φ ∈ L2; ∥φ∥m,s = ∥(1 + x2)s/2(1 - Δ)m/2φ∥L2 < ∞} is the weighted Sobolev space. We also show the existence of the usual scattering states. Our result for n = 3, 4 is an improvement of Hayashi and Hirata, Nonlinear Anal. 31 (1998), 671-685.

Original languageEnglish
Pages (from-to)133-147
Number of pages15
JournalAsymptotic Analysis
Volume21
Issue number2
Publication statusPublished - 1999 Oct 1
Externally publishedYes

Keywords

  • Derivative nonlinear Schrödinger equations
  • General space dimensions
  • Global small solutions

ASJC Scopus subject areas

  • Mathematics(all)

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