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Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation

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  • 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 $\psi(0,x)=\psi_0(x)$.
    Mathematics Subject Classification: Primary: 35A05, 35A15; Secondary: 35Q51, 35Q55.


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