## 3 Deterministic Single Soliton Generation  
To demonstrate the deterministic access of single DKS in the four previously proposed configurations of \(a\) and \(b\), we conducted 200 repetitions of the LLE simulation, employing distinct initial random fields for each setting. In the case of configuration \((a_{1},b_{1})\), we maintained a constant normalized power \((\left|S\right|^{2})\) of 1.04. The normalized detuning \((\Delta)\) was swept from \(-1\) to 0.98 over a simulation period of 3 \(\upmu\)s, as depicted in Figure 4a. For parameters \((a_{2},b_{2})\), the magnitude \(\left|S\right|^{2}\) was set to 1.1538, and the detuning \(\Delta\) ranged from \(-1\) to 1.5918. Similarly, for \((a_{3},b_{3})\) parameters, \(\left|S\right|^{2}\) equaled 1.08, with \(\Delta\) spanning from \(-1\) to 1.24. Finally, in the case of \((a_{4},b_{4})\) parameters, \(\left|S\right|^{2}\) was fixed at 1.09, and \(\Delta\) varied from \(-1\) to 0.93. The values of \(\Delta\) and \(\left|S\right|^{2}\) chosen for DKS generation along each path are determined by the trajectories indicated by the magenta arrows in Figure 3.  
We emphasize that these trajectories do not cross the chaotic region. Figure 4a-d show that we obtain a single DKS state; see the temporal shape (center) and spectrum (right). It is noteworthy that when the value of \(a\) is \(a=2\pi\times 7.71\) [GHz], the contour of the spectrum deviates from the a hyperbolic-secant shape. However, increasing \(a\) causes this shape to align more closely with the sech(x) shape. Note that configuration \((a_{1},b_{1})\) does not conform to the sech(x) shape, unlike the other configurations. Figure 4a-d show that \(U_{\text{intra}}\) remains stable and noise-free along the selected paths, supporting controlled DKS generation. The absence of abrupt fluctuations confirms that the trajectory avoids chaotic regions, essential for stability. Additionally, the convergence of \(U_{\text{intra}}\) to a constant value indicates that the DKS reaches a stable configuration, which is crucial for maintaining coherence in different applications. Figure 5 shows the number of peaks (\(N_{\text{peaks}}\)) for the 200 simulations in the four proposed \(a\) and \(b\) configurations. We obtained in a single peak in all the realizations, indicating that a single DKS was always generated, demonstrating that the paths are deterministic. Although these configurations enable the generation of a single DKS, these simulations do not demonstrate that the DKS is stable. For this reason, we perform a stability analysis. To validate DKS generation under variations in the DCM caused by changes in aspect ratio and manufacturing tolerances, which affect the radius, loss, coupling coefficient, and dispersion of the device, we conducted a Monte Carlo simulation. In this simulation, we varied the radius, loss, coupling coefficient, and dispersion randomly within the ranges specified in Table 1. This approach allowed us to evaluate the robustness of DKS generation under realistic manufacturing fluctuations, ensuring reliable device performance within practical tolerance levels. Figure 4: **(a)** OFC features for \((a_{1},b_{1})\). **(b)** OFC features for \((a_{2},b_{2})\), **(c)** OFC features for \((a_{3},b_{3})\). **(d)** OFC features for \((a_{4},b_{4})\). The features are the total intracavity energy—Untra vs. slow time **(left)**, optical intensity **(center)**, optical spectrum **(right)**. As shown in Figure 6, we conducted 200 Monte Carlo iterations across the four paths defined in Figure 4, achieving DKS generation in 175 cases (Figure 6a), 130 cases (Figure 6b), 160 cases (Figure 6c), and 181 cases (Figure 6d). These results indicate a DKS generation probability exceeding 80% for Figure 6a,c,d and 65% for Figure 6b. While variations in the aspect ratio influenced the results, the method still showed robustness and adaptability to specific structural changes.