Nonlinear Dispersive Equations on Modulation Spaces by NLS Approximations: Numerical Methods and Analysis of Nonlinear Dispersive Equations in Quantum Mechanics and Nonlinear Optics

Abstract

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. By using tools of time-frequency analysis, we obtain some improved local well-posedness results for the NLS, equations with data in modulation spaces. This is the first time in which particle methods are being used for solving such equations. We numerically test our new method for a variety of linear and nonlinear problems. In particular we are interested in nonlinear equations which generate structures that have non-smooth fronts. It is remarkable to see that our particle method is capable of capturing the nonlinear regime of a compacton-cotnpacton type interaction.