# Modeling Optical Technologies with Continuous and Discrete Nonlinear Schrödinger Equations [Elektronische Ressource] / Christopher Chong. Betreuer: G. Schneider

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Modeling Optical Technologies with Continuousand Discrete Nonlinear Schr¨odinger EquationsZur Erlangung des akademischen Grades einesDOKTORS DER NATURWISSENSCHAFTENvon der Fakult¨at fu¨r Mathematikder Universit¨at Karlsruhe (TH) genehmigteDISSERTATIONvonMSc.-Math. Christopher ChongausBoston, MA USATag der mu¨ndlichen Pru¨fung: 23. Juli 2009Referent: Prof. Dr. Guido SchneiderKorreferent: Prof. Dr. Michael PlumThis thesis is dedicated to my late uncle Bill.ContentsAcknowledgements vii1 Introduction 12 Interaction of modulated pulses in the nonlinear Klein-Gordon equation 52.1 Pulse interaction approximated by the NLS equation . . . . . . . . . . . . . . 62.1.1 Modulated pulse deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Single pulse approximation . . . . . . . . . . . . . . . . . . . . . . . . 92.1.3 Two pulse approximation . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.4 Justiﬁcation of the two pulse approximation . . . . . . . . . . . . . . . 132.2 Numerical integrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.1 Leapfrog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.2 Method of lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.3 Implicit pseudo-spectral method . . . . . . . . . . . . . . . . . . . . . 182.2.4 Geometric pseudo-spectral method . . . . . . . . . . . . . . . . . . . . 202.2.5 Comparison of methods . . . . . . . . . . . . .

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