The singular perturbation limit of a variational problem from nonlinear fiber optics

Abstract

Among the fiber optical models the dispersion managed (DM) systems have attracted interest due to their favorable properties compared to the more traditional approaches which rely on classical solitons. Numerical and physical experiments have shown that in the limit of strong dispersion management, there exist stable nearly periodic pulses. It was later found that these breathing pulses correspond to the ground states solutions of a certain averaged DM problem with positive and even zero residual dispersion. It was not clear, however, how DM solitons in the zero residual dispersion regime are related to those ones in the positive residual dispersion regime. In this paper, DM solitons in the mean-zero case are obtained as DM solitons in the limit of vanishing positive residual dispersion. This corresponds to studying the singular perturbation limit of energy functionals of the form ϕ (ε)(u)=(ε/2)∫ R |u′| 2 dx−F(u) with a highly nonlocal operator F.