Perturbed dissipative solitons: A variational approach

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 twophoton 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.

Numerical realization of the variational method for generating self-trapped beams

“…Generic results can be adequately presented for propagation constant λ = 1/2 in Eq. (10) and values of parameters s = 0.05 and k = 5, p = 30 in Lagrangian terms (15) and (16), respectively, unless specified otherwise.…”

“…The VA makes it possible to predict nonlinear modes with different degrees of accuracy, depending on the choice of the underlying ansatz [12], in diverse nonlinear systems. These include, in particular, photonic lattices in nonlinear media, induced by non-diffracting beams [13], dissipative media [14,15], Bose-Einstein condensates [16,17], parity-time-symmetric lattices [18], dynamics of spatio-temporal solitons in a periodic medium [19], and even the prediction of complex hopfion modes with two topological charges (twisted vortex ring) [20]. The VA was also applied to nonlocal GNLSEs [21,22].…”

Free-carrier-assisted mid-infrared microcavity soliton generation

“…An extended Lugiato-Lefever equation (LLE) considering both the MPA and FC effects is written in the form of the Landau-Euler equation to derive the approximate theoretical solution of solitons, 29 δL δA…”

Variational approach to study soliton dynamics in a passive fiber loop resonator with coherently driven phase-modulated external field

“…Moreover, it helps to achieve greater control over the generated CS. On the other hand, a theoretical tool depending on the variational approach was first introduced by Anderson [16] for nonlinear pulse propagation, since then it has been used extensively in conservative systems [17][18][19] as well as in dissipative systems including active [20,21] and passive resonators [22][23][24] to study the * maitrayee@iitkgp.ac.in pulse dynamics. In the first part of our study, we made a detailed analysis of the dynamics of a single CS in the framework of variational treatment and obtained several closed form expressions of the CS characteristics which can be determined according to the phase profile of the CW.…”