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Asymptotics in time for nonlinear nonlocal Schrödinger equations with a source. (English) Zbl 0927.35109

Summary: We consider the asymptotic behavior of the solution of the Cauchy problem for the nonlinear nonlocal Schrödinger equation (NNS) with a source \[ iu_t+| u|^2u+ iKu= f(x,t),\quad t>0,\quad x\in\mathbb{R}, \]
\[ u(x,0)= \overline u(x). \] Here the linear pseudodifferential operator \(Ku\) is defined by \[ Ku={1\over 2\pi} \int_{\mathbb{R}} e^{ipx}K(p)\widehat u(p,t)dp. \] The source in the NNS equation makes essential alterations to the asymptotic behavior. We study the cases of both small and large initial data.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
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