Influence of intrapulse Raman scattering on stationary pulses in the presence of linear and nonlinear gain as well as spectral filtering

We examine numerically the influence of intrapulse Raman scattering (IRS) on the stable stationary pulses in the presence of constant linear and nonlinear gain as well as spectral filtering. Numerical results show that the small change of the value of the parameter describing IRS leads to qualitatively different behavior of the evolution of pulse amplitudes. We prove that the strong dependence of the pulse dynamics on the parameter describing IRS is related to the existence of the Poincaré-Andronov-Hopf bifurcation and the appearance of the unstable limit cycle.

Figure 1

Evolution of the pulse amplitude $\eta \left(x\right)$ with distance according to Eq. (1) with $\delta =-0.02,\beta =0.05,\varepsilon =0.1$, and $\mu =-0.02$ for (a) $\gamma =0.0540833$ and $x_{max}=300$, with initial conditions ${\eta }_0=1.25$ and $k_0=-0.8$; (b) γ  =  0.064 083 3, $x_{max}=300$ with initial conditions ${\eta }_0=1.25$ and $k_0=-0.8$; and (c) γ  =  0.074 083 3, $x_{max}=80$ with initial conditions ${\eta }_0=1.48$ and $k_0=-0.8$.

Figure 2

Graphs of $\alpha =\alpha \left(\gamma \right)$ (solid line) and $\Delta =\Delta \left(\gamma \right)$ (dashed line).

Figure 3

Phase portrait of the ODE system (3) for $\delta =-0.02,\beta =0.05,\varepsilon =0.1,\mu =-0.02,\gamma =0.0540833$, and initial conditions ${\eta }_0=1.25$ and $k_0=-0.8$ (marked with a small circle) for $x_{max}=500$, demonstrating the existence of the stable focal point (marked with a large circle).

Figure 4

Phase portrait of the system of ODEs (3) for $\delta =-0.02,\beta =0.05,\varepsilon =0.1,\mu =-0.02,\gamma =0.0640833$, and initial conditions ${\eta }_0=1.25$ and $k_0=-0.8$ (marked with a small circle) for $x_{max}=500$, demonstrating the appearance of the limit cycle. The fixed point is marked with a large circle.

Figure 5

Phase portrait of the ODE system (3) for $\delta =-0.02,\beta =0.05,\varepsilon =0.1,\mu =-0.02,\gamma =0.0740833$, and initial conditions ${\eta }_0=1.09$ and $k_0=-0.7$ (marked with a small circle) for $x_{max}=120$, demonstrating the existence of unstable focal point (marked with a large circle).

Figure 6

Evolution of the pulse amplitude $\eta \left(x\right)$ with $\delta =-0.02,\beta =0.05,\varepsilon =0.1$, and $\mu =-0.02$ for $\gamma =0.0540833$ and $x_{max}=300$, with initial conditions ${\eta }_0=1.25$ and $k_0=-0.8$. The solid line denotes the numerical solution of the CCQGLE (1) and the circles denote the numerical solution of the ODE system (3).