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 language | English |
|---|---|
| Pages (from-to) | 1387-1409 |
| Number of pages | 23 |
| Journal | Nonlinearity |
| Volume | 9 |
| Issue number | 6 |
| DOIs | |
| Publication status | Published - 1996 Dec 1 |
| Externally published | Yes |
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)
- Applied Mathematics