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Elsevier, Physica D: Nonlinear Phenomena, (370), p. 29-39

DOI: 10.1016/j.physd.2018.01.002

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On the classification of the spectrally stable standing waves of the Hartree problem

Journal article published in 2017 by Vladimir Georgiev ORCID, Atanas Stefanov
This paper is available in a repository.
This paper is available in a repository.

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Abstract

We consider the fractional Hartree model, with general power non-linearity and space dimension. We construct variationally the "normalized" solutions for the corresponding Choquard-Pekar model - in particular a number of key properties, like smoothness and bell-shapedness are established. As a consequence of the construction, we show that these solitons are spectrally stable as solutions to the time-dependent Hartree model. In addition, we analyze the spectral stability of the Moroz-Van Schaftingen solitons of the classical Hartree problem, in any dimensions and power non-linearity. A full classification is obtained, the main conclusion of which is that only and exactly the "normalized" solutions (which exist only in a portion of the range) are spectrally stable.