Wave envelopes with second-order spatiotemporal dispersion. II. Modulational instabilities and dark Kerr solitons

A simple scalar model for describing spatiotemporal dispersion of pulses, beyond the classic “slowly varying envelopes $+$ Galilean boost” approach, is studied. The governing equation has a cubic nonlinearity, and we focus here mainly on contexts with normal group-velocity dispersion. A complete analysis of continuous waves is reported, including their dispersion relations and modulational instability characteristics. We also present a detailed derivation of exact analytical dark solitons, obtained by combining direct-integration methods with geometrical transformations. Classic results from conventional pulse theory are recovered asymptotically from the spatiotemporal formulation. Numerical simulations test theoretical predictions for modulational instability and examine the robustness of spatiotemporal dark solitons against perturbations to their local pulse shapes.

Figure 1

Dispersion relations for cw solutions with κ $=$ $+$10${}^{-3}$, ${\rho }_0$ $=$ 1.0, and α $=$ 1.0. (a) Ellipse when $s$ $=$ $+$1 [representative of regimes with sgn($s\kappa$) $=$ $+$1] and (b) hyperbole when $s$ $=$ $-$1 [representative of regimes with sgn($s\kappa$) $=$ $-$1]. The families of curves are centered on (Ω,$K$) $=$ ($s\alpha ,0$), so the dominant effect of varying α is to translate the curves along the Ω axis. Blue-upper curves: forward wave. Red-lower curves: backward wave.

Figure 2

Dispersion relation [obtained by solving Eq. (6) numerically] for Fourier mode (5) when κ $=$ $+$10${}^{-3}$, ${\rho }_0$ $=$ 1.0, α $=$ 1.0, and $s$ $=$ $+$1. Note that there are always four branches because of the quartic nature of Eq. (6). (a) The propagation constant of the perturbation (real part of the complex wave number $K_p$) comprises two ellipses. (b) The spatial growth rate of the mode (imaginary part of $K_p$) has a short-wave MI at high ${\Omega }_p$. Inset: Bow-tie-type structure of the classic long-wave MI curve (the additional two roots are zero). These small-scale features in Λ${}_p$(${\Omega }_p$) are not apparent from visual inspection of the larger-scale plot.

Figure 3

Spontaneous development of modulational instability in the anomalous-GVD regime ($s$ $=$ $+$1) when κ $=$ $+$10${}^{-3}$, α $=$ 1.0, and ${\rho }_0$ $=$ 1.0. In this simulation, the level of noise is set to 0.01$\%$ (ε $=$ 10${}^{-4}$). The zero-frequency spectral component [i.e., the dominant peak in the fast Fourier transform (FFT) that is associated with the Ω $=$ 0 cw background field] has been filtered from the dataset. The first sidebands that start to grow are centered on the most unstable frequency ${\Omega }_{p0}\simeq {\left(2{\rho }_0\right)}^{1/2}$ as predicted by linear analysis. The results are qualitatively unchanged for κ $=$ $-$10${}^{-3}$ because the long-wave instability region is independent of κ.

Figure 4

Schematic illustrating the geometry of solution (17) and, in particular, the distinction between pulses propagating in the forward (FWD) and backward (BWD) longitudinal directions. In the (τ,ζ) plane, the minimum of the gray dip travels along the lines τ $-$ $V_0\zeta$ $=$ 0 and τ $+$ $V_0\zeta$ $=$ 0, respectively, where the intrinsic velocity $V_0$ is given by Eq. (16a). The trajectory makes an angle Ξ relative to the longitudinal axis where tan Ξ $=$ $V_0$.

Figure 5

Schematic illustrating the distinction between forward-propagating black and gray dark solitons. The black soliton (those solutions with $F$ $=$ 0) has an intrinsic velocity $V_{0\mathrm{b}}$ [given by Eq. (16b)], whereas, its more general gray counterpart (here, a solution with $F>0$) has an intrinsic velocity $V_0$ [where $V_0(F>0)>V_{0\mathrm{b}}$].

Figure 6

Self-reshaping of perturbed black ($F$ $=$ 0) pulses (28) toward an exact soliton with (a) κ $=$ $+$10${}^{-3}$ (hyperbolic scenario) and (b) κ $=$ $-$10${}^{-3}$ (elliptic scenario—note that the qualitative features of these curves are similar to those in Ref. 23 for Helmholtz spatial dark solitons). Solid bars denote theoretical predictions of the asymptotic full width $\Delta {\tau }_{\infty }$. Other parameters: ${\rho }_0=1.0$ and α $=$ 1.0; note that the cw background of each pulse has no long-wave MI since $s$ $=$ $-$1.