Nonlinear Interaction of Pulses [Elektronische Ressource] / Martina Chirilus-Bruckner. Betreuer: G. Schneider

.Nonlinear Interaction of PulsesZur Erlangung des akademischen Grades einesDOKTORS DER NATURWISSENSCHAFTENvon der Fakultät für Mathematik der Universität Karlsruhe (TH) genehmigteDISSERTATIONvonDipl.-Math. techn. Martina Chirilus-Bruckneraus Vatra Dornei, RumänienTag der mündlichen Prüfung: 23. Juli 2009Referent: Prof. Dr. Guido SchneiderKorreferent: Prof. Dr. Michael PlumIntroductionConsider the initial value problem for a nonlinear wave equation, say, the cubic Klein-Gordon equation2 2 3∂ u =∂ u−u+u ,t xwhere x,t,u =u(x,t)∈R, with the initial profile (as depicted in Fig. 0.1) given byu(x,t)| =u (x,t)| ,t=0 pulse t=0∂ u(x,t)| =∂ u (x,t)|t t=0 t pulse t=0with2 ik x+iω t 2 ik x+iω t1 1 2 2u (x,t) =εA (ε(x−c t),ε t)e +εA (ε(x−c t+d),ε t)e +c.c.,pulse 1 1 2 2where c ,k ,ω ,d∈R,0<ε≪ 1, and A (,t) is a localized function for all t∈R.j j j ju(,t)|t=0 c2c1xFigure 0.1: Initial profile given by two localized structures traveling with different velocitiesNow, how does this initial profile evolve in time? How do the localized structures interact? Will acollision destroy them? What if they travel with the same speed? What if we take more than two?Thepresentworkanalyzesallthesenaturalquestionsinthecontextofvariousnonlinearwaveequationswhich arise in applications such as nonlinear fiber optics and photonics, where the localized structuresrepresent light pulses which can be used to transport and process digital data (see Ch. 7).