Pulse-train propagation in nonlinear Kerr media governed by higher-order dispersion

TL;DR

This work addresses pulse propagation in Kerr media described by an extended NLSE including second-, third-, and fourth-order dispersion. Using a traveling-wave ansatz, the authors derive an amplitude equation and construct three analytic pulse-train solutions, each expressed as a product of Jacobi elliptic functions, with explicit dependence on the dispersion coefficients and a shared propagation velocity. In the long-wave limit, these trains degenerate to quartic and dipole solitons, and their amplitude–duration relations are governed by the sign of $\Omega=30\epsilon/\gamma$. Numerical simulations demonstrate stable propagation over multiple dispersion lengths in realistic materials, indicating potential for experimental generation and slow-light control in optical fibers and waveguides.

Abstract

We discover three novel classes of pulse-train waveforms in an optical Kerr nonlinear medium possessing all orders of dispersion up to the fourth order. We show that both single- and double humped pulse-trains can be formed in the nonlinear medium. A distinguishing property is that these structures have different amplitudes, widths and wavenumbers but equal velocity which depends on the three dispersion parameters. More importantly, we find that the relation between the amplitude and duration of all the newly obtained pulse-trains is determined by the sign of a joint parameter solely. The results show that those optical waves are general, in the sense that no specified conditions on the material parameters are assumed. Considering the long-wave limit, the derived pulse-trains degenerate to soliton pulses of the quartic and dipole kinds.

Figures (7)

• Figure 1: Intensity profiles of pulse-train solutions with parameters $\gamma =2$, $k=0.9$, $\xi _{0}=0$; (a) pulse-train solution (\ref{['16']}) when $\alpha =-1,$$\sigma =-0.25,$$\epsilon =-0.25$; (b) pulse-train solution (\ref{['25']}) when $\alpha =0.49,$$\sigma =1,$$\epsilon =0.5$; (c) pulse-train solution (\ref{['34']}) when $\alpha =0.49,$$\sigma =1,$$\epsilon =0.5.$
• Figure 2: Intensity profiles of pulse-train solutions with parameter $k=0.9$; (a) pulse-train solution (\ref{['16']}), (b) pulse-train solution (\ref{['25']}), (c) pulse-train solution (\ref{['34']}). Other parameters are the same as given in Fig. 1.
• Figure 3: Evolution of pulse-train solutions with parameter $k=0.999$; (a) pulse-train solution (\ref{['16']}), (b) pulse-train solution ( \ref{['25']}), (c) pulse-train solution (\ref{['34']}). Other parameters are the same as given in Fig. 1.
• Figure 4: Evolution of pulse-train solutions with parameter $k=0.999$; (a) pulse-train solution (\ref{['16']}), (b) pulse-train solution ( \ref{['25']}), (c) pulse-train solution (\ref{['34']}). Other parameters are the same as given in Fig. 1.
• Figure 5: Dependence of the velocity of pulse-trains $v$ on the group velocity dispersion parameter $\alpha$ for $\sigma =1$ and different values of $\epsilon$; $\epsilon =-0.25$ (thick line), $\epsilon =-0.3$ (dashed line) and $\epsilon =-0.35$ (dotted line).