Open AccessOn the focusing generalized Hartree equation
Author(s) -
Anudeep Kumar Arora,
Svetlana Roudenko,
Kai Yang
Publication year - 2020
Publication title -
mathematics in applied sciences and engineering
Language(s) - English
Resource type - Journals
ISSN - 2563-1926
DOI - 10.5206/mase/10855
Subject(s) - hartree , convolution (computer science) , nonlinear system , nonlinear schrödinger equation , mathematics , mathematical analysis , type (biology) , schrödinger equation , scattering , dynamics (music) , mathematical physics , physics , quantum mechanics , computer science , ecology , machine learning , artificial neural network , biology , acoustics
In this paper we give a review of the recent progress on the focusing generalized Hartree equation, which is a nonlinear Schrodinger-type equation with the nonlocal nonlinearity, expressed as a convolution with the Riesz potential. We describe the local well-posedness in H1 and Hs settings, discuss the extension to the global existence and scattering, or finite time blow-up. We point out different techniques used to obtain the above results, and then show the numerical investigations of the stable blow-up in the L2 -critical setting. We finish by showing known analytical results about the stable blow-up dynamics in the L2 -critical setting.