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Local existence in time of solutions to the elliptic-hyperbolic Davey-Stewartson system without smallness condition on the data. (English) Zbl 0907.35120

Summary: We study the initial value problem for the elliptic-hyperbolic Davey-Stewartson systems \[ i\partial_tu +\Delta u=c_1 | u|^2 u+c_2u \partial_{x_1} \varphi, \quad (\partial^2_{x_1} -\partial^2_{x_2}) \varphi= \partial_{x_1} | u|^2, \quad u(0,x)= \varphi (x), \tag{1} \] where \(\Delta= \partial^2_{x_1} +\partial^2_{x_2}\), \(c_1,c_2\in \mathbb{R}\), \(u\) is a complex valued function and \(\varphi\) is a real valued function. When \((c_1,c_2) =(-1,2)\) the system (1) is called DSI equation in the inverse scattering literature. Our purpose in this paper is to prove the local existence of a unique solution to (1) in the Sobolev space \(H^2 (\mathbb{R}^2)\) without the smallness condition on the data which was assumed in previous works. Our result is a positive answer to Question 7 in [C. E. Kenig, G. Ponce and L. Vega, Contemp. Math. 189, 353-367 (1995; Zbl 0846.35128)].

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)

Citations:

Zbl 0846.35128
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References:

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