Exact analytical solutions for the variational equations derived from the nonlinear Schrödinger equation

Abstract

By means of the variational formalism for the nonlinear Schrödinger equation, we find an explicit relation for the power of a pulse in terms of its duration, chirp and fiber parameters (group-velocity dispersion and self-phase modulation parameters). Then, using that relation, we derive the explicit analytical expressions for the variational equations corresponding to the amplitude, width, and chirp of the pulse. The derivation of the analytical expressions for the variational equations is possible for the condition when the Hamiltonian of the system is zero. Finally, for Gaussian and hyperbolic secant ansatz, we show good agreement between the results obtained from the analytical expressions and the direct numerical simulation of the nonlinear Schrödinger equation.

Dispersion-Managed Optical Fiber Systems with Zero Hamiltonian

“…In particular, for Gaussian and hyperbolic secant shaped pulses, we have showed a good agreement between the analytical results and the full numerical simulation of the NLSE for a set of parameters. In this work we numerically generalize the results obtained in reference [12] by considering a large range of fiber and pulse parameters. Then we investigate the effectiveness of using Gaussian and hyperbolic secant ansätse in our exact analytical solutions.…”

“…After some algebraic manipulations [12], we derive the following exact analytical expressions for the pulse amplitude, width and chirp in a dispersive and nonlinear optical fiber medium, with respect to the GVD ) To check the validity of the above derived pulse parameters equations, we compare the results obtained from the numerical solution of the NLSE (1) and the analytical expression for nonlinear pulse propagation in a single-mode fiber. Figures (1) and (2) are the results obtained respectively for the Gaussian and hyperbolic secant shaped pulses.…”

“…Mikhailov has reported the validity of the variational analysis [11]. Very recently, we have derived the exact analytical expressions for the variational equations corresponding to the amplitude, width and chirp under the condition when the Hamiltonian of the system is zero [12]. These results have been obtained by assuming a generalized ansatz function for the shape of a pulse propagating in a nonlinear fiber with anomalous dispersion.…”