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Zbl 0910.35113
Hayashi, Nakao; Hirata, Hitoshi
Local existence in time of small solutions to the elliptic-hyperbolic Davey-Stewartson system in the usual Sobolev space. (English)
[J] Proc. Edinb. Math. Soc., II. Ser. 40, No.3, 563-581 (1997). ISSN 0013-0915; ISSN 1464-3839/e

Summary: We study the initial value problem to the Davey-Stewartson system $$i \partial_t u+c_0 \partial^2_{x_1} u+\partial^2_{x_2} u=c_1| u |^2 u+c_2u \partial_{x_1} \varphi, \quad (x,t)\in \bbfR^2 \times\bbfR$$ $$\partial^2_{x_1} \varphi+ c_3 \partial^2_{x_2} \varphi = \partial_{x_1} | u|^2, \quad u(x,0) =\varphi (x),$$ for the elliptic-hyperbolic case in the usual Sobolev space. We prove local existence and uniqueness in $H^{5/2}$ with a condition such that the $L^2$ norm of the data is sufficiently small.

MSC 2000:: *35Q53 KdV-like equations
35A07 Local existence and uniqueness theorems (PDE)
76B15 Wave motions (fluid mechanics)

Keywords: smallness condition; initial value problem to the Davey-Stewartson system; elliptic-hyperbolic case; Sobolev space

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