Dark-bright quadratic solitons with a focusing effective Kerr nonlinearity

Dark solitons are traditionally considered to exist in defocusing Kerr nonlinearity media. We investigate dark quadratic solitons with a focusing effective Kerr nonlinearity and a sine-oscillatory nonlocal response. A nonlinear refractive index with a focusing sine-oscillatory response leads to a defocusing effect with a strong degree of nonlocality, which causes the formation of dark solitons. By analyzing the modulational instability, we determine the parameter domain for dark quadratic solitons with a stable background and numerically obtain dark-bright soliton solutions in the form of pairs, which avoid radiative phenomena. Based on a numerical simulation, we find that all dark-bright soliton pairs are unstable after a relatively long propagation distance, and their stabilities are affected by the soliton interval and the degree of nonlocality.

Figure 1

Sketch map of the nonlinear refractive index induced by (a) a Gaussian-form beam and (b) a tanh-form beam in the strongly (solid red lines) and weakly (dashed blue lines) nonlocal regimes. Dotted lines denote the intensity of the beams.

Figure 2

Spectrum of the MI gain for the case $r=-s=+1$ at different values of $\sigma$: (a) $\sigma =1$ and (b) $\sigma =1/2$.

Figure 3

Intensity profiles of dark soliton pairs for $r=-s=+1$ at (a) $\alpha =0.1$ and $l_n=29.8$, (b) $\alpha =0.1$ and $l_n=49.7$, (c) $\alpha =0.1$ and $l_n=69.5$, and (d) $\alpha =0.5$ and $l_n=13.3$. Solid lines denote the FW component; dashed lines, the SH component.

Figure 4

(a) Peak and dip separations $w_s$ of a single soliton for the FW and SH components versus the degree of nonlocality $\alpha$; (b) peak intensity $I_m$ of solitons versus degree of nonlocality $\alpha$. Solid lines denote the FW component; dashed lines, the SH component.

Figure 5

Simulation of the FW (left) and SH (right) components of dark-bright soliton pairs. (a, b) $\alpha =0.01, l_n=94.2$; (c, d) $\alpha =0.01, l_n=597$; (e, f) $\alpha =0.01, l_n=785$; (g, h) $\alpha =0.02, l_n=422$.

Figure 6

Simulation of the (a, c) FW and (b, d) SH components of dark-bright soliton pairs with or without MI. $\sigma =1/4$ (a, b); $\sigma =1$ (c, d). For all $\alpha =0.02, l_n=111$.