Perturbed dissipative solitons: A variational approach

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 into 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 semianalytic treatment is useful for gaining physical insight into complex soliton-evolution processes.

Figure 1

(a)–(c) Temporal and (d)–(f) spectral evolution of an ODS in three cases under a single perturbation: (a) and (d) IRS acts alone with ${\tau }_R=0.1$, (b) and (e) self-steepening acts alone with $s=0.1$, and (c) and (f) FCD acts alone with $\theta =0.0044$. The other parameters used in the simulations are $K=0.01$, $g_0=0.01$, $g_2=0.01$, and $\alpha =0$. The input (dotted trace) and output pulse shapes are also shown on the top of each panel.

Figure 2

(a) Temporal position ${\tau }_p$ and (b) frequency shift $\mathrm{\Omega }$ as a function of distance in four different cases listed at the bottom. Changes in the peak intensity ${\left|u\right|}_0^2$ and pulse width ${\tau }_w$ in the same four cases are shown in (c) and (d), respectively. The solid red line, dashed blue line, dot-dashed green line, and solid gray line show the variational predictions for ${\tau }_R$, $\theta$, $s$, and all three parameters, respectively, whereas red circles, blue diamonds, green triangles, and gray squares represent the corresponding numerical data. The other parameters used in the simulations are $K=0.01$, $g_0=0.01$, $g_2=0.01$, and $\alpha =0$.

Figure 3

(a) Changes in pulse energy as a function of $\xi$ in the four cases of Fig. 2. (b) Changes in the IRS-induced redshift and FCD-induced blueshift as predicted by the approximated analytic expressions. The blue (red) dot-dashed and dashed lines represent the cases when the initial values of the pulse parameters and the average values of the pulse parameters are considered for the free-carrier (Raman) case, respectively.

Figure 4

(a) Comparison of output intensity profile at $\xi =200$ between full numerical simulation (red dots) and variational prediction (blue trace). (b) Frequency shift as a function of distance for ${\tau }_R=0.1$. Blue circles and red squares correspond to sech-shape pulse and ODS inputs, respectively. The solid green line shows variational prediction of the frequency shift. The parameters used in the simulations are $K=0.01$, $g_0=0.01$, $g_2=0.0001$, and $\alpha =0$.

Figure 5

(a) Temporal evolution of ODS under the TOD perturbation. Input (dotted trace) and output pulse shapes are shown on top. (b) Changes in ODS position with distance as predicted by the variational technique (solid blue line) are compared with numerical data (red circles). The parameters used in the simulations are $K=0.01$, $g_0=0.001$, $g_2=0.001$, and ${\delta }_3=0.1$.