Dark spatial solitary waves in a cubic-quintic-septimal nonlinear medium

We consider the evolution of light beams in nonlinear media exhibiting nonlinearities up to the seventh order wherein the beam propagation is governed by the cubic-quintic-septimal nonlinear Schrödinger equation. An exact analytic solution that describes dark solitary wave propagation is obtained, based on a special ansatz. Unlike the well-known $\text{tanh}$-profile dark soliton in Kerr media, the present one has a functional form given in terms of “${\text{sech}}^{2/3}$”. The requirements concerning the optical material parameters for the existence of this localized structure are discussed. This propagating solitary wave exists due to a balance among diffraction, cubic, quintic, and septimal nonlinearities. We also investigate its stability under initial pertubations by employing numerical simulations. The modulational stability of the continuous-wave background, which supports the dark soliton, is also studied analytically. Collisions between stable dark solitons are finally investigated.

Figure 1

Plots showing the characteristic features of MI for ${\alpha }_1=1$. (a): Evolution of the upper limit of the gain spectrum, $k_c$ versus ${\alpha }_2$, for different values of ${\alpha }_3$. (b): Peak value of the gain, $G_{MI}$ versus ${\alpha }_2$. (c): Power level of the cw background, $P_0$ versus ${\alpha }_2$. (d): Septimal nonlinear term, ${\alpha }_4$ versus ${\alpha }_2$.

Figure 2

Plot showing the evolution of the spatial profile of the soliton, as a function of the propagation distance $z$, for ${\alpha }_1=1$, ${\alpha }_3=1$, and ${\alpha }_2=1.55$, 0.35, and 0.17. The value of ${\alpha }_4$ is given by the parametric conditions (22). The other parameter values used here are: (a) ${\alpha }_4=0.2294$, $\mathrm{\Omega }=-0.2$, $\kappa =-0.8508$; (b) ${\alpha }_4=1.0159$, $\mathrm{\Omega }=-0.2$, $\kappa =-0.0813$; (c) ${\alpha }_4=2.0915$, $\mathrm{\Omega }=-0.2$, $\kappa =-0.0498$. The initial condition of the simulation is the soliton profile given by formula (23a).

Figure 3

Plot showing the evolution of the spatial profile of the soliton, as a function of the propagation distance $z$, for ${\alpha }_1=1$, ${\alpha }_2=1$, and ${\alpha }_3=0.026$, 0.5, and 1.8. The value of ${\alpha }_4$ is given by the parametric conditions (22). The other parameter values used here are: (a) ${\alpha }_4=2.4036\times {10}^{-4}$, $\mathrm{\Omega }=-0.2$, $\kappa =-13.0208$; (b) ${\alpha }_4=0.0889$, $\mathrm{\Omega }=-0.2$, $\kappa =-0.7150$; (c) ${\alpha }_4=1.1520$, $\mathrm{\Omega }=-0.2$, $\kappa =-0.2275$. The initial condition of the simulation is the soliton profile given by formula(23a).

Figure 4

Stability diagram in the space of nonlinearity parameters. (a) : Plot of the stability parameter $\xi$ as a function of (${\alpha }_2$, ${\alpha }_3$). (b): Plot of $1/G_{MI}$. Here $\mathrm{\Omega }=-0.2$ and the value of $\kappa$ is given by parametric conditions (22).

Figure 5

Plot showing the evolution of the spatial profile of the soliton, as a function of the propagation distance $z$, in the presence of a strong perturbation generated by a photon noise having an average power of $2.7\times {10}^{-5}$. Here ${\alpha }_1=1$, ${\alpha }_2=0.18$, ${\alpha }_3=1.8$, ${\alpha }_4=6.4$, $\mathrm{\Omega }=-0.2$, and $\kappa =-0.0461.$ The initial condition of the simulation is the soliton profile given by the formula (23a).

Figure 6

Plot showing the evolution of the intensity profile of two colliding solitons, as a function of the propagation distance $z$. The initial condition of the simulation is given by the formula (30), with ${\alpha }_1=5$, ${\alpha }_2=0.18$, ${\alpha }_3=1.8$, ${\alpha }_4=6.4$, ${\mathrm{\Omega }}_1=-{\mathrm{\Omega }}_2=2$, and ${\kappa }_1={\kappa }_2=-20$.