The Cauchy problem for the elliptic–hyperbolic Davey–Stewartson system in Sobolev space

We study the initial value problem for the elliptic–hyperbolic Davey–Stewartson systemsequation(0.1){i∂tu+Δu=c1|u|2u+c2u∂x1φ,(t,x)∈R3,(∂x12−∂x22)φ=∂x1|u|2,u(0,x)=ϕ(x), where Δ=∂x12+∂x22, c1,c2∈Rc1,c2∈R, u is a complex valued function and φ is a real valued function. Our purpose is to prove the local existence and uniqueness of the solution for (0.1) in the Sobolev space H3/2+(R2)H3/2+(R2) with small mass. Our methods rely heavily on Hayashi and Hirata (1996) [11], but we improve partial results of it, which got global existence of small solutions to (0.1) in weighted Sobolev space H3,0∩H0,3H3,0∩H0,3. Our main new tools are Kenig–Ponce–Vega type commutator estimate in Kenig, Ponce and Vega (1993) [16] and its variant form.