# Perturbed Dissipative Solitons: A Variational Approach

## TL;DR

This paper uses a variational method with a Pereira--Stenflo ansatz to analyze the dynamics of perturbed dissipative solitons governed by a Ginzburg--Landau equation, providing insights and accurate predictions for complex behaviors.

## Contribution

It introduces a variational approach with an exact unperturbed soliton solution to model and analyze perturbed dissipative solitons in active waveguides, offering new analytical and numerical insights.

## Key findings

- Derived six coupled differential equations for soliton parameters.
- Predicted spectral and temporal shifts accurately compared to numerical simulations.
- Provided simple analytic expressions for soliton shifts.

## Abstract

We adopt a variational technique to study the dynamics of perturbed dissipative solitons, whose evolution is governed by a Ginzburg--Landau equation (GLE). As a specific example of such solitons, we consider a silicon-based active waveguide in which free carriers are generated through two-photon absorption. In this case, dissipative solitons are perturbed by physical processes such as third-order dispersion, intrapulse Raman scattering, self-steepening, and free-carrier generation. To solve the variational problem, we adopt the Pereira--Stenflo soliton as an ansatz since this soliton is the exact solution of the unperturbed GLE. With this ansatz, we derive a set of six coupled differential equations exhibiting the dynamics of various pulse parameters. This set of equations provides considerable physical insight in the complex behavior of perturbed dissipative solitons. Its predictions are found to be in good agreement with direct numerical simulations of the GLE. More specifically, the spectral and temporal shifts of the chirped soliton induced by free carriers and intrapulse Raman scattering are predicted quite accurately. We also provide simple analytic expressions of these shifts by making suitable approximations. Our semi-analytic treatment is useful for gaining physical insight into complex soliton-evolution processes.