## 2 Characterization of Pump Power and Detuning Parameter Space
The location of this region in the parameter space poses challenges for accessing DKS. The difficulty occurs because the path must avoid the chaotic zone, requiring the proposal of a non-linear route. Next, we simulated the DCM by introducing the mode interaction with the parameters \((a,b)\) we previously proposed. The shift of the DKS area to the left and bottom of the region where combs are generated is apparent. This new position of the single soliton region makes it accessible in a path at constant pump power and with detuning that sweeps directly from blue to red, as shown by the black arrow in Figure 1f. We next study the impact of parameters \((a,b)\) on the generation of single DKS. Instead of fixed values, we aim to identify effective combinations of \(a\) and \(b\), increasing the probability of single DKS generation. This approach deepens system understanding and helps pattern recognition, aiding future research. To define \((a,b)\), we analyzed the single DKS generation probability at four key points in the parameter space \((\Delta,|S|^{2})\), producing an explicit function of \(a\) and \(b\). The choice of these specific points was deliberate, guided by their strategic location below the chaotic region, avoiding crossing this region and making it accessible in a path with detuning that sweeps directly from blue to red. The probability calculation at each point was determined by systematically varying the \((a,b)\) values across the specified ranges, where \(a\) ranged from 0 to \(2\pi\times 18\) [GHz], and b ranged from 0 to 20. The number of iterations used in the probability calculation was 30 simulations, introducing variability in the initial field and in the integration path through Gaussian noise with \(\sigma_{\text{noise}}=10^{-9}\,[\text{W}^{1/2}]\). Figure 2 presents the probability of single DKS generation as a function of different parameter values \((a,b)\), exploring these variations at the four specified points. Notably, various configurations of \((a,b)\) exist, enabling reliable single DKS generation. However, many of these configurations are unattainable in a physics device, so we prefer those with a value of \(a\) less than \(a=2\pi\times 10\) [GHz]. To maximize the generation probability, we have deliberately chosen specific values corresponding to the regions of the graph marked with red, magenta, green, and black rectangles. These values, proven to be potential in terms of DKS generation, are as follows: \((a_{1}=2\pi\times 7.71\) [GHz], \(b_{1}=10.56)\), \((a_{2}=2\pi\times 8.67\) [GHz], \(b_{2}=16.32)\), \((a_{3}=2\pi\times 9.71\) [GHz], \(b_{3}=13.44)\), \((a_{4}=2\pi\times 15.71\) [GHz], \(b_{4}=15.48)\). This selection is based on the maximum generation probability associated with the chosen points, aiming to ensure reliable device performance under consideration. \[t_{R}\frac{\partial E(t,\tau)}{\partial t}=\left[-\alpha-i\delta_{0}-iL\frac{ \beta_{2}}{2}\frac{\partial^{2}}{\partial\tau^{2}}+i\gamma L|E|^{2}\right]E+ \sqrt{\phi}E_{\text{in}}+i\sum_{\mu}\frac{a/2}{\mu-b}\tilde{E}_{\mu}(t)e^{i\mu\tau} \tag{1}\]1 (a) DKS magenta, green, and black rectangles probability generation for the point \((\Delta_{1}=0.92\), \(|S_{1}|^{2}=1.05)\). (b) DKS probability generation for the point \((\Delta_{2}=1,|S_{2}|^{2}=1)\). (c) DKS probability generation for the point \((\Delta_{3}=1.58\), \(|S_{3}|^{2}=1.31)\). (d) DKS probability generation for the point \((\Delta_{4}=1.79,|S_{4}|^{2}=1.79)\). Figure 3 displays the parameter space \((\Delta,|S|^{2})\) for the previously proposed parameters \((a,b)\). It is important to note that all four figures show a single DKS area in the lower right part of the parameter space, highlighted in black. This signifies direct access to the soliton region without passing through the chaotic zone. In Figure 3d, the \((\Delta,|S|^{2})\) parameter space exhibits a larger DKS region than with other choices of \(a\) and \(b\) (see Figure 3).