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Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. (English) Zbl 1247.35143

Summary: We study the Cauchy problem for the focusing time-dependent Schrödinger-Hartree equation \[ i \partial_t \psi + \triangle \psi = -({|x|^{-(n-2)}}\ast |\psi|^{\alpha})|\psi|^{\alpha - 2} \psi, \quad \alpha\geq 2, \] for space dimension \(n \geq 3\). We prove the existence of solitary wave solutions and give conditions for formation of singularities in dependence of the values of \(\alpha\geq 2\) and the initial data \(\psi(0,x)=\psi_0(x)\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A15 Variational methods applied to PDEs
35Q51 Soliton equations
35C08 Soliton solutions
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