A Study on Nonlinear Dispersive Partial Differential Equations by RDT Method: Exploring Numerical Methods for Nonlinear Dispersive Equations in Quantum Mechanics and Nonlinear Optics

Abstract

The nonlinear dispersive 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.The aim of this paper is to propose and analyze various numerical methods for some representative classes of nonlinear dispersive 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 dispersive equations, it is believed that the methods and results achieved for these equations can be applied or extended to more general cases.