## 3 Dark Solitons
### Black Solitons under Dissipation  
We next study the dynamics of the black solitons under the presence of the radiation and viscothermal losses. The effective NLS (18), which includes a linear loss term, can be studied via the direct perturbation theory for dark solitons [33]. The loss term does not vanish for \(\tau_{1}\rightarrow\infty\) and, thus, in this limit (where \(\partial^{2}A/\partial\tau_{1}^{2}\to 0\)), the evolution of the black soliton's background \(A\equiv\eta(\chi_{2})\) is determined by the equation:  
\[i\frac{\partial\eta}{\partial\chi_{2}}-q\left|\eta\right|^{2}\eta=-i\Lambda\eta. \tag{28}\]  
Solutions of Equation (28) can be sought in the form \(\eta(\chi_{2})=\eta_{0}(\chi_{2})\exp[i\phi(\chi_{2})]\), where \(\eta_{0}(\chi_{2})\) describes the loss-induced change of the background amplitude, and \(\phi(\chi_{2})\) is the spatially varying phase of the background. Substituting the above ansatz into Equation (28), and separating real and imaginary parts, we derive the equation for the amplitude of the background as follows:  
\[\frac{\partial\eta_{0}}{\partial\chi_{2}}=-\Lambda\eta_{0}. \tag{29}\]  
Integrating Equation (29), we find that the background amplitude decays according to the following exponential law:  
\[\eta_{0}(\chi_{2})=\eta_{0}(0)\exp(-\Lambda\chi_{2}). \tag{30}\]  
In terms of the original space coordinate, the amplitude of the background of the black soliton decreases exponentially as:  
\[\eta_{0}(x)=\eta_{0}(0)\exp\left(-\frac{x}{L_{l}}\right), \tag{31}\]  
where  
\[L_{l}=\frac{c_{0}}{\Lambda e^{2}\omega_{B}} \tag{32}\]  
Figure 3: (color online) (**a**) a contour plot of black soliton of the form of Equation (25), obtained by numerically integrating the lossless version of Equation (3) (\(R_{\omega}=0,R_{H}=0\)) with \(\epsilon=0.029\) (\(2\epsilon\eta R_{0}=8816\) Pa) (\(170\) dB) and carrier frequency \(f=600\) Hz; (**b**) numerical spatial profile of black soliton calculated at \(t=1.41\) s (blue line). Green dashed lines are the corresponding analytical envelope result of Equation (25). The light green thick line denotes the nonlinear length \(L_{NL}\) and dispersion length \(L_{D}\), with \(L_{NL}=L_{D}=21.55\) m; (**c**) a numerical contour plot for dispersive effect, obtained by numerically integrating the lossless version of Equation (3) (\(R_{\omega}=0,R_{H}=0\)). Keeping the same width, we only decrease the amplitude of the driver to \(\epsilon=0.0095\) (\(2\epsilon\eta R_{0}=2888\) Pa) (\(160\) dB) such as the nonlinearity can not balance the dispersion; (**d**) numerical spatial profile of dispersive wave calculated at \(t=1.41\) s.  
is the dissipation length. Thus, the envelope of the approximate black soliton solution reads  
\[\begin{split}& p_{e}(x,t)\approx 2\epsilon P_{0}\eta_{0}(0)e^{ \frac{-\Lambda e^{2}\omega_{B}}{\epsilon_{0}}x}\\ &\times\tanh\left[\epsilon\eta_{0}e^{\frac{-\Lambda e^{2}\omega_{ B}}{\epsilon_{0}}x}\sqrt{\left|\frac{q}{k_{r}^{\prime}}\right|}\omega_{B}\left(t- \frac{k_{r}^{\prime}}{c_{0}}x\right)\right].\end{split} \tag{33}\]  
During propagation, the amplitude of background \(\eta\) decreases (due to the presence of loss), while the minimum intensity is always zero. The black soliton does not move against the background, and the velocity of the background only depends on the carrier frequency. Thus, the linear loss only affects the black soliton's background amplitude and width.