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On the quadratic nonlinear Schrödinger equation in three space dimensions. (English) Zbl 1004.35112

From the introduction: We study the Cauchy problem for the quadratic nonlinear Schrödinger equation in three space dimensions: \[ iu_t+\frac 12\Delta u={\mathcal N}(u),\;x\in\mathbb{R}^3,\;t\in\mathbb{R},\qquad u(0,x)=u_0, \quad x\in\mathbb{R}^3,\tag{1} \] where the nonlinear term is \({\mathcal N}(u)=\lambda_1 u^2+\lambda_2 \overline u^2\), \(\lambda_1,\lambda_2 \in \mathbb{C}\). The aim of this paper is to prove global existence in time of small solutions to the Cauchy problem (1) and to construct the usual scattering states.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35K15 Initial value problems for second-order parabolic equations
35P25 Scattering theory for PDEs
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