Nonlinear Dispersive Equations on Modulation Spaces By Nls Approximations: Numerical Methods for Nonlinear Dispersive Equations in Quantum Mechanics and Nonlinear Optics

Abstract

The aim of this paper is to propose and analyze variousnumerical methods for some representative classes of nonlinear dispersiveequations, which mainly arise in the problems of quantum mechanics andnonlinear optics. Extensive numerical results are also reported, which aregeared towards demonstrating the efficiency and accuracy of the methods, aswell as illustrating the numerical analysis and applications. By using tools oftime-frequency analysis, we obtain some improved local well-posedness resultsfor the NLS, equations with data in modulation spaces. This is the first timein which particle methods are being used for solving such equations. Wenumerically test our new method for a variety of linear and nonlinear problems.In particular we are interested in nonlinear equations which generatestructures that have non-smooth fronts. It is remarkable to see that our particlemethod is capable of capturing the nonlinearregime of a compacton-cotnpacton type interaction.