Global existence and asymptotic behaviour in time of small solutions to the elliptic-hyperbolic Davey-Stewartson system

Nakao Hayashi, Hitoshi Hirata

Research output: Contribution to journalArticlepeer-review

26 Citations (Scopus)

Abstract

We study the initial value problem for the elliptic-hyperbolic Davey-Stewartson system {i∂tu + △u = c1\u\2u + c2u∂xiφ (t, x) ∈ ℝ3 (∂x12 - ∂x22)φ = ∂x1|u|2 u = u(t, x) φ = φ(t, x) u(0, x) = φ(x) where △ = ∂x12 + ∂x22, c1, c2 ∈ ℝ, u is a complex valued function and φ is a real valued function. When (c1,c2) = (-1, 2) the above system is called a DSI equation in the inverse scattering literature. Our purpose in this paper is to prove global existence of small solutions to this system in the usual weighted Sobolev space H3,0 ∩ H0,3, where Hm,l = {f ∈ L2; ∥(1 - ∂x12 - ∂x22)m/2(1 + x12 + x22)l/2f∥L2 < ∞}. Furthermore, we prove L∞ time decay estimates of solutions to the system such that ∥u(t)∥L∞ ≤ C(1 + |t|)-1.

Original languageEnglish
Pages (from-to)1387-1409
Number of pages23
JournalNonlinearity
Volume9
Issue number6
DOIs
Publication statusPublished - 1996 Dec 1
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

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