Description of the suppression of the soliton self-frequency shift by bandwidth-limited amplification

A perturbation study of the suppression of the soliton self-frequency shift by the bandwidth-limited optical amplification is proposed. The stability of the equilibrium point for the soliton amplitude and velocity identified by the adiabatic approximation of the soliton perturbation theory (SPT) is analyzed by a numerical solution of a linearized system in the neighborhood of the equilibrium point. The obtained analytical expressions for the eigenvalues of the linearized system allow the determination of the values of pulse and material parameters for which the equilibrium point is stable. A perturbation approach that leads to the research of the equation of strongly nonlinear Duffing-Van der Pol oscillator is suggested. The last equation is explored by two different methods. First, the recently obtained results for this equation by the hyperbolic perturbation method are used. Next, the hyperbolic Lindstedt-Poincare perturbation method is applied to the exploration of this equation. The equilibrium velocity of the perturbed stationary solution was calculated as a critical value of the control parameter in both methods. It turned out that the coupling of the equilibrium velocity and the amplitude of the perturbed stationary solution in both methods is similar to the relation between the soliton amplitude and velocity derived by the adiabatic approximation of SPT. The change in the form of the perturbed stationary solution has also been identified by means of the hyperbolic Lindstedt-Poincare perturbation method.