The nonlinear Schrödinger equation as a macroscopic limit for an oscillator chain with cubic nonlinearities

Johannes Giannoulis, Alexander Mielke
2003 Nonlinearity  
We consider the nonlinear model of an infinite oscillator chain embedded in a background field. We start from an appropriate modulation ansatz of the space-time periodic solutions to the linearized (microscopic) model and derive formally the associated (macroscopic) modulation equation, which turns out to be the nonlinear Schrödinger equation. Then we justify this necessary condition rigorously for the case of nonlinearities with cubic leading terms; i.e. we show that solutions that have the
more » ... ns that have the form of the assumed ansatz for t = 0 preserve this form over time-intervals with a positive macroscopic length. Finally, we transfer this result to the analogous case of a finite but large periodic chain and illustrate it by a numerical example. Mathematics Subject Classification: 37K60, 70F45, 70K70, 34E13, 35Q55
doi:10.1088/0951-7715/17/2/011 fatcat:jpkh6lqfqzfytm353w44wdao74