Generalized sixth-order dispersion solitons

Encouraged by recent experimental breakthroughs, we investigate the types of bright solitons that can exist in the presence of a Kerr nonlinearity and quadratic, quartic, and sextic dispersion by examining the nature of the soliton tails. We verify the soliton shape by full numerical calculations and find analytic solutions for some parameters. We reveal the sensitivity of the interaction dynamics to the particular choice of the dispersion parameters. These results provide a framework for experiments in this regime and for exploring parameter spaces with even higher dimensions.

Figure 1

Schematic of the four root configurations of Eq. (3), indicating the nature of the tail of the soliton solutions. Here ${\beta }_6=-1{{\mathrm{ps}}^6\mathrm{mm}}^{-1}, {\beta }_4=-2{{\mathrm{ps}}^4\mathrm{mm}}^{-1}$, and $\mu =1{\mathrm{mm}}^{-1}$. (a) ${\beta }_2=2{{\mathrm{ps}}^2\mathrm{mm}}^{-1}$ (Configuration A); (b) ${\beta }_2=0{{\mathrm{ps}}^2\mathrm{mm}}^{-1}$ (Configuration B); (c) ${\beta }_2=-2{{\mathrm{ps}}^2\mathrm{mm}}^{-1}$ (Configuration C); and (d) ${\beta }_2=-4{{\mathrm{ps}}^2\mathrm{mm}}^{-1}$ (Configuration D). Each is indicated by green dots in Fig. 3. Features of the configurations are summarized in Table 1.

Figure 2

Cross section through parameter space for ${\beta }_4=0$ and ${\beta }_6=-1{{\mathrm{ps}}^6\mathrm{mm}}^{-1}$. The colors correspond to root configurations given in Table 1. The boundaries ${\mu }_0$ and ${\mu }_c$ are given in Eq. (5) and (6), respectively.

Figure 3

Cross sections taken through the general parameter space with ${\beta }_6=-1{{\mathrm{ps}}^6\mathrm{mm}}^{-1}$. (a) $\mu =1{\mathrm{mm}}^{-1}$; (b) ${\beta }_4=-1.3{{\mathrm{ps}}^4\mathrm{mm}}^{-1}$. The colors correspond to the regions given in Table 1. (Yellow dotted line) ${\beta }_2=0$; (purple dotted line) ${\beta }_4=0$; (dark blue and orange curves) ${\mu }_0^{\pm }$ [Eq. (7)]; (red curve) ${\mu }_c$ [Eq. (8)]; (dark green dotted line) cross section of ${\beta }_4=-1.3{{\mathrm{ps}}^4\mathrm{mm}}^{-1}$ in (a) corresponding to (b); (pink crosses) hyperbolic secant exact solution discussed in Sec. 4b; (light green line) A, B, C, and D correspond to the root configurations in Fig. 1.

Figure 4

Soliton amplitude versus time for ${\beta }_6=-1{\mathrm{ps}}^6{\mathrm{mm}}^{-1}, \gamma =1W^{-1}{\mathrm{mm}}^{-1}$ and $\mu =1{\mathrm{mm}}^{-1}$ for (a) solution with exponential tails with ${\beta }_2=-3{{\mathrm{ps}}^2\mathrm{mm}}^{-1}, {\beta }_4=-4{{\mathrm{ps}}^4\mathrm{mm}}^{-1}$ on a linear scale, and (b) on a logarithmic scale. (c) and (d) Same as (a) and (b) but for a solution with oscillatory tails with ${\beta }_2=0.5{{\mathrm{ps}}^2\mathrm{mm}}^{-1}, {\beta }_4=-4{{\mathrm{ps}}^4\mathrm{mm}}^{-1}$. (e) and (f) Solution close to boundary ${\mu }_c$ with ${\beta }_2=-0.34{{\mathrm{ps}}^2\mathrm{mm}}^{-1}, {\beta }_4=-0.1{{\mathrm{ps}}^4\mathrm{mm}}^{-1}$. (g) and (h) Exact analytic solution (purple crosses in Fig. 3) with ${\beta }_2=-0.8518{{\mathrm{ps}}^2\mathrm{mm}}^{-1}, {\beta }_4=-1.11415{{\mathrm{ps}}^4\mathrm{mm}}^{-1}$.

Figure 5

Propagation of Gaussian pulses separated by $8\mathrm{ps}$ for parameters shown by the green dots in Fig. 3, corresponding to Configurations (a) A, (b) B, (c) C, and (d) D. The dispersion parameters are given for the corresponding configurations in Fig. 1.

Figure 6

As Fig. 5, but with the Gaussian pulses separated by $10\mathrm{ps}$.

Figure 7

Cross section through parameter space for ${\beta }_6=-1{{\mathrm{ps}}^6\mathrm{mm}}^{-1}$ and ${\beta }_2=0$. The colors correspond to regions given in Table 1 (Regions A and B).

Figure 8

Cross section through the parameter space for small, positive $\mu$ and ${\beta }_6=-1{{\mathrm{ps}}^6\mathrm{mm}}^{-1}$. The colors correspond to regions given in Table 1 (Regions A, C, and D). The blue curve bounding Region D is given analytically by ${\beta }_2=-8{\beta }_4^2/5{\beta }_6$.