On the Dispersion-Managed Optical Fiber Systems with Zero Hamiltonian

The dynamics of nonlinear pulse propagation in optical fibers is governed by the famous nonlinear Schrödinger equation (NLSE), in which the group-velocity dispersion (GVD) and self-phase modulation (SPM) form a basic set of optical processes describing a broad range of realistic physical situations. The NLSE is studied extensively in order to understand the influence of combining those effects. A particular initial condition of the pulse leads to a particular dynamical process during its propagation in a nonlinear and dispersive optical medium. The more famous of such a process is the conventional soliton which can be observed when the effect of anomalous GVD is exactly balanced by the SPM in optical fibers. Thus formed soliton pulse can then propagate without any deformation of its shape. Under special cases, the NLSE is completely integrable and the corresponding soliton solutions can be derived using the standard technique called inverse scattering transform. But the family of NLSE equations governing most practical cases like conventional fiber transmission system, dispersion-managed (DM) fiber system are not completely integrable in general. Even though some perturbation methods were reported to investigate the behaviour of physically interesting non-integrable NLSE family, researchers working in nonlinear optics and other fields mostly rely on numerical methods and Lagrangian variational method to study the system dynamics. Variational method is one of the widely used approximation techniques which has been applied to study the dynamics of various pulse parameters with respect to the fiber parameters, to estimate the pulse-to-pulse interaction length and to find the fixed point solutions of the DM fiber systems. In this work, by means of variational formalism for the NLSE, we derive exact analytical expressions for the variational equations corresponding to the amplitude, width and chirp of the pulse in terms of initial pulse parameters, fiber parameters and the distance of propagation of the pulse; under the condition when the Hamiltonian of the system is zero. Then, for Gaussian and hyperbolic secant ansatz, we check the validity of the obtained analytical results to describe pulse propagation in optical fiber in presence of high order effects. As a practical application of our results, we consider the design of the DM fiber systems and we derive an analytical expression for the Gordon-Haus jitter.