A study of nonlinear relativistic Hartree equation

Abstract

The aim of this paper is to propose and analyze various numerical methods for some representative classes of nonlinear Hartree equations, which mainly arise in the problems of quantum mechanics and nonlinear optics. Extensive numerical results are also reported, which are geared towards demonstrating the efficiency and accuracy of the methods, as well as illustrating the numerical analysis and applications. Although the subjects considered here is merely a small sample of nonlinear Hartree equations, it is believed that the methods and results achieved for these Hartree equations can be applied or extended to more general cases. The nonlinear Hartree equations, including a large body of classes, are widely used models for a great number of problems in the fields of physics, chemistry and biology, and have gained a surge of attention from mathematicians ever since they were derived. In addition to mathematical analysis, the numeric of these equations is also a beautiful world and the studies on it have never stopped.