Numerical Methods and Their Analysis for Some Nonlinear Dispersive Equations

Abstract

This thesis is devoted to the numerical solutions of several classes of dispersive equations, namely, the coupled Schrodinger-Poisson (SP) type system, the Klein-Gordon (KG) equation, the sine-Gordon (SG) and perturbed Schrodinger (PNLS) equations. For SP type system, various numerical methods were compared for ground states calculation and dynamics, in both nonrelativistic and relativistic cases. Also, simplification in spherical symmetry case was considered. For the KG equation, special effort was made to the nonrelativistic limit regime involving a small parameter \epsilon, where solutions propagate with a wavelength O(\epsilon^2) in time. Rigorous error estimates show that classical finite difference time integrators require time step to be O(\epsilon^3). A Gautschi-type integrator was proposed, and time step restriction is relaxed to O(\epsilon^2) in nonlinear case and O(1) in linear case. In the last topic, SG and PNLS were compared for modeling 2D light bullets, with the help of efficient numerical methods. Again, the numerical methods applied here were rigorously analyzed. For all these subjects, extensive numerical examples were reported to illustrate and support the results.