## 3 Dark Solitons
### Black Solitons  
We start with the black soliton, with \(\phi=0\), which at the NLS level is expressed as:  
\[A=\eta_{0}\text{tanh}\left(\eta_{0}\sqrt{\left|\frac{q}{k_{r}^{\prime\prime}} \right|}\tau_{1}\right)\exp\left(-iq\eta_{0}^{2}\chi_{2}\right). \tag{23}\]  
The corresponding approximate black soliton solution of Equation (6) is as follows:  
\[\begin{split} P(\chi,\tau)\approx& 2\eta_{0}\text{tanh}\left[ \epsilon\eta_{0}\sqrt{\left|\frac{q}{k_{r}^{\prime\prime}}\right|}\left(\tau-k _{r}^{\prime}\chi\right)\right]\\ &\times\cos\left(\omega\tau-k_{r}\chi-\epsilon^{2}q\eta_{0}^{2} \chi\right),\end{split} \tag{24}\]  
which is a function of parameters \(\chi\) and \(\tau\). In the original coordinates, space \(x\) and time \(t\), the approximate black soliton solution for the pressure \(p\) reads  
\[\begin{split} p(x,t)&\approx 2\epsilon\eta_{0}P_{0}\text{tanh}\left[ \epsilon\eta_{0}\sqrt{\left|\frac{q}{k_{r}^{\prime\prime}}\right|}\omega_{B} \left(t-\frac{k_{r}^{\prime}}{c_{0}}x\right)\right]\\ &\times\cos\left(\omega\omega_{B}t-\frac{k_{r}\omega_{B}}{c_{0}} x-q\epsilon^{2}\eta_{0}^{2}\frac{\omega_{B}}{c_{0}}x\right),\end{split} \tag{25}\]  
where \(k_{r}^{\prime}/c_{0}\) is the inverse group velocity at the carrier frequency that is independent of the amplitude of the background \(2\epsilon\eta_{0}P_{0}\).  
To ensure the balance between nonlinearity and dispersion, we should also pay attention to the dispersion length \(L_{D}\) and the nonlinearity length \(L_{NL}\) providing the length scales over which dispersive or nonlinear effects become important for pulse evolution. These lengths are expressed as follows [14]:  
\[L_{D}=\frac{t_{0}^{2}\omega_{B}c_{0}}{\left|k_{r}^{\prime\prime}\right|}, \tag{26}\]  
and  
\[L_{NL}=\frac{P_{0}^{2}c_{0}}{\left|q\right|\omega_{B}A_{0}^{2}}, \tag{27}\]  
where \(t_{0}\) and \(A_{0}\) are the characteristic width and the amplitude of the propagating envelope, respectively. From the black soliton solution, Equation (25), we find \(t_{0}=\left(\epsilon\eta_{0}\sqrt{\left|q/k_{r}^{\prime\prime}\right|}\omega_{ B}\right)^{-1}\) and \(A_{0}=\epsilon\eta_{0}P_{0}\). Substituting them in Equations (26) and (27), we obtain \(L_{NL}/L_{D}\sim 1\).  
Now, we study numerically the evolution of the approximate black soliton solution of Equation (25) in the lattice. We start by integrating the lossless version of Equation (3) with a driver of the form given by Equation (25) at \(x=0\) with \(\epsilon=0.029\) (\(2\epsilon\eta P_{0}=8816\) Pa) (\(170\) dB) and carrier frequency \(f=600\) Hz. The numerical results are shown in Figure 3a,b. We observe that the black soliton with zero intensity at the center propagates with a constant velocity, amplitude and width as shown in the contour plot (Figure 3a). Good agreement between the simulations (blue line) calculated at \(t=1.41\) s and the corresponding analytical solution of Equation (25) (green dashed line) is shown in Figure 3b. In order to confirm that the black solitons are due to the balance between dispersion and nonlinearity, we also compare Figure 3a,b to the unbalanced case where the boundary condition has the same width but a smaller amplitude, \(\epsilon=0.0095\) (\(2\epsilon\eta P_{0}=2888\) Pa) (\(160\) dB), shown in Figure 3c,d. In Figure 3d, it is observed that the soliton is not formed and the initial wavepacket spreads.