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

Hristo Genev; George Venkov

doi:10.3934/dcdss.2012.5.903

Article Contents

2012, Volume 5, Issue 5: 903-923. Doi: 10.3934/dcdss.2012.5.903

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

Hristo Genev^1, and
George Venkov^2,

1.
Soﬁa University St. Kl. Ohridski, Faculty of Mathematics and Informatics
2.
Technical University of Soﬁa, Faculty of Applied Mathematics and Informatics

Received: December 2010

Revised: June 2011

Published: October 2012

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Abstract

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 $\psi(0,x)=\psi_0(x)$.

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Mathematics Subject Classification: Primary: 35A05, 35A15; Secondary: 35Q51, 35Q55.

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