A comprehensive analysis of the propagation of short pulses in fiber nonlinear directional couplers is presented. In particular, the limitations as well as the merits of the variational approach to describe power- and phase-controlled pulse switching are discussed. Relying on beam propagation calculations a trial function that accounts for variable width, amplitude, phase, and chirp of the pulses is proposed. The corresponding Euler-Lagrangian equations for the pulse parameters are derived and solved. Concerning power-controlled switching an excellent agreement of the switching curve with beam propagation results is found. Moreover, even the evolution of the various pulse parameters is shown to be described reasonably by the analytical model used. Eventually, optimum criteria for efficient phase-controlled switching may be easily derived by using the model presented.

The generalization of the adiabatic perturbation approach of $N$-soliton interactions in the nonlinear Schrödinger equation (NLSE) has been shown to lead to the complex Toda lattice with $N$ nodes. This allows us to predict the asymptotic behavior of trains of $N$ solitonlike pulses with approximately equal amplitudes and velocities, but with arbitrary phase differences. These predictions agree very well with the numerical results.

An analytical description of the soliton self-frequency shift (SSFS) of dark solitons is presented, using the conservation laws of the nonlinear Schrödinger equation. It is shown that the influence of intrapulse Raman scattering on dark solitons crucially depends on the sign of soliton’s velocity. Our analytical expressions qualitatively explain the main features of the SSFS of the dark solitons observed recently both in experiment [A. M. Weiner, R. N. Thurston, W. J. Tomlinson, J. P. Heritage, D. E. Leaird, E. M. Kirschner, and R. J. Hawkins, Opt. Lett. 14, 868 (1989)] and in numerical simulation [A. M. Weiner, R. N. Thurston, W. J. Tomlinson, J. P. Heritage, D. E. Leaird, E. M. Kirschner, and R. J. Hawkins, Opt. Lett. 14, 868 (1989); Yu. S. Kivshar and V. V. Afanasjev, ibid. 16, 285 (1991)].

In this paper we study the transitions of stationary to pulsating solutions in the complex cubic-quintic Ginzburg-Landau equation (CCQGLE) under the influence of nonlinear gain, its saturation, and higher-order effects: self-steepening, third-order of dispersion, and intrapulse Raman scattering in the anomalous dispersion region. The variation method and the method of moments are applied in order to obtain the dynamic models with finite degrees of freedom for the description of stationary and pulsating solutions. Having applied the first model and its bifurcation analysis we have discovered the existence of families of subcritical Poincaré-Andronov-Hopf bifurcations due to the intrapulse Raman scattering, as well as some small nonlinear gain and the saturation of the nonlinear gain. A phenomenon of nonlinear stability has been studied and it has been shown that long living pulsating solutions with relatively small fluctuations of amplitude and frequencies exist at the bifurcation point. The numerical analysis of the second model has revealed the existence of Poincaré-Andronov-Hopf bifurcations of Raman dissipative soliton under the influence of the self-steepening effect and large nonlinear gain. All our theoretical predictions have been confirmed by the direct numerical solution of the full perturbed CCQGLE. The detailed comparison between the results obtained by both dynamic models and the direct numerical solution of the perturbed CCQGLE has proved the applicability of the proposed models in the investigation of the solutions of the perturbed CCQGLE.

A method for the description of the N-soliton interaction, which generalizes in a natural way the Karpman-Solov'ev one for the nonlinear Schrödinger (NLS) equation, is proposed. Using it, we derive a nonlinear system of equations describing the dynamics of the parameters of N well separated solitons with nearly equal amplitudes and velocities. Next we study an exhaustive list of perturbations, relevant for nonlinear optics, which include linear and nonlinear dispersive and dissipative terms, effects of sliding filters, amplitude and phase modulation, etc. We prove that the linear perturbations affect each of the solitons separately, while the nonlinear ones also lead to additional interactive terms between neighboring solitons. Under certain approximations we show that the N-soliton interaction for the unperturbed NLS equation is described by the complex Toda chain (CTC) with N nodes, which is a completely integrable dynamical system with 2N degrees of freedom. A comparison made by numeric simulation shows that CTC gives an adequate description for the soliton interactions for a number of choices of the initial conditions.