Soliton dynamics in optical fibers using the generalized traveling-wave method

The generalized traveling wave method (GTWM) is applied to the nonlinear Schrödinger (NLS) equation with general perturbations in order to obtain the equations of motion for an ansatz with six collective coordinates, namely the soliton position, the amplitude, the inverse of the soliton width, the velocity, the chirp, and the phase. The advantage of the new ansatz is that it yields three pairs of canonically conjugated coordinates and momenta that all are well-behaved. The new ansatz is applied to model the dynamics of a soliton in a dispersion-shifted optical fiber described by the generalized NLS, including dissipation, higher-order dispersion, Raman scattering, and self-steepening perturbations. It is shown that the GTWM is equivalent to the modified method of moments, which considers the time variation of the norm, the first and the second moment of the norm, the momentum, the first moment of the momentum, and the energy for the perturbed NLS equation.

Figure 1

$A, \zeta , \xi , \eta , C$, and $\mathrm{\Phi }$ vs. propagation distance $t$ for a large propagation distance.

Figure 2

$E_p, {\mathrm{\Omega }}_p, T_p$, and the normalized chirp parameter vs. propagation distance $t$ for a larger fiber length. For the normalized chirp (lower right panel), an old definition $C_p/N_p$ (blue curve) displays growing oscillations, while the new definition $C/N$ (black curve) results in more subdued oscillations. Factors $N_p={10}^{-2}$ and $N={10}^{24}{\mathrm{s}}^{-2}$ are used to display the curves on comparable vertical scales.