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Remarks on nonlinear Schrödinger equations in one space dimension. (English) Zbl 0803.35137

Summary: We consider the initial value problem for nonlinear Schrödinger equations, \[ i \partial_ tu + {1 \over 2} \partial^ 2u = F(u, \partial u, \overline u, \partial \overline u), \quad (t,x) \in \mathbb{R}^ + \times \mathbb{R}, \qquad u (0,x) = u_ 0(x), \quad x \in \mathbb{R}, \tag{1} \] where \(\partial = \partial_ x = \partial/ \partial x\) and \(F:\mathbb{C}^ 4 \to \mathbb{C}\) is a polynomial having neither constant nor linear terms. Without a smallness condition on the data \(u_ 0\), it is shown that (1) has a unique local solution in time if \(u_ 0\) is in \(H^{3,0} \cap H^{2,1}\), where \[ H^{m,s} = \biggl\{ f \in S' : \| f \|_{m,s} = \bigl \| (1 + x^ 2)^{s/2} (1 - \Delta)^{m/2} f \bigr \|_ 2 < \infty \biggr\},\;m, s \in \mathbb{R}. \]

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q60 PDEs in connection with optics and electromagnetic theory
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
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