The stabilization of the nonlinear model arising in pulse propagation in optical fiber

Abstract

We analyze the stabilization of the third order nonlinear Schrödinger equation in a bounded interval under the effect of a localized damping mechanism. That is, we consider the following equation: iu t + αu xx + iβu xxx + |u| 2 u + ia(x)u = 0. Where u = u(x, t) is a complex valued function defined in (0, L) × (0, +∞) and α, β are real constants.This equation models a pulse propagation in a long-distance and high-speed optical fiber transmission system. Using multiplier techniques and a special uniform continuation theorem we prove the exponential decay of the total energy associated with the above system.