## 2 Characterization of Pump Power and Detuning Parameter Space
One standard way to normalize Equation (1) is to let  
\[\Delta=\frac{\delta_{0}}{\alpha}\qquad S=E_{\text{in}}\sqrt{\frac{\gamma L\phi} {\alpha^{3}}} \tag{4}\]  
\[T=\frac{\alpha t}{t_{R}}\qquad\psi=\sqrt{\frac{\gamma L}{\alpha}}E \tag{5}\]  
\[\beta=\frac{8\pi^{3}R}{t_{R}^{2}\alpha}\beta_{2}=\frac{4\pi^{2}L}{t_{R}^{2} \alpha}\beta_{2} \tag{6}\]  
\[\theta=\frac{2\pi}{T_{R}}\tau \tag{7}\]  
in which case Equation (1) becomes  
\[\frac{\partial\psi}{\partial T}=-i\Bigg{[}\frac{8\pi^{3}R}{t_{R}^{2}}\frac{ \beta_{2}}{2\alpha}\Bigg{]}\frac{\partial^{2}\psi}{\partial\theta^{2}}+i|\psi |^{2}\psi-(i\Delta+1)\psi+S+i\frac{1}{\alpha}\sum_{\mu}\left[\frac{a/2}{\mu-b} \widetilde{\psi}_{\mu}(T)e^{i\mu b}\right] \tag{8}\]  
where  
\[t_{R}\frac{\partial E(t,\tau)}{\partial t}=\left[-\alpha-i\delta_{0}-iL\frac{ \beta_{2}}{2}\frac{\partial^{2}}{\partial\tau^{2}}+i\gamma L|E|^{2}\right]E+ \sqrt{\phi}E_{\text{in}}+i\sum_{\mu}\frac{a/2}{\mu-b}\tilde{E}_{\mu}(t)e^{i\mu\tau} \tag{1}\]0  
To solve this equation numerically, we used the split-step Fourier method. Note that the normalized parameter definitions proposed in Equation (8) facilitate our analyses in a more general framework, and allow for comparison with other methods presented in the literature [8; 15]. To perform the simulation of the LLE model, we used the same parameters as in [8] which correspond to a Si\({}_{3}\)N\({}_{4}\) structure. The parameters used were a main radius of 100 \(\upmu\)m, \(t_{R}=1/226\) GHz, \(\beta_{2}=-4.7\times 10^{-26}\) [s\({}^{2}\)m\({}^{-1}\)], \(\alpha=1.61\times 10^{-3}\), \(\gamma=1.09\) [W\({}^{-1}\)m\({}^{-1}\)], \(L=2\pi\times 100\) [\(\upmu\)m], and \(\theta=6.4\times 10^{-4}\). The parameters chosen in the LLE model are founded on experimentally validated values from real physical devices [8; 32; 33]. Additionally, we replicated real conditions in the simulation by initializing the intracavity field with Gaussian noise, set at a standard deviation \(\sigma_{\text{noise}}=10^{-9}\) [W\({}^{1/2}\)], which is added at each iteration. This approach ensures that our simulations are closely aligned with practical implementations, accurately reflecting the behavior of physical photonic systems. Due to the complexity and computational cost, thermal shift was not included in the model. Additionally, we set the pump power for the single MRR at a detuning (\(\Delta=0\)) and a pump power of \(\left|P_{\text{in}}\right|^{2}=6\) (57.1 mW), ensuring a well-defined behavior in the cnoidal wave region. Similarly, for the DCM, we tuned the pump power at a detuning (\(\Delta=0\)) and \(\left|P_{\text{in}}\right|^{2}=5\) (47.6 mW). These values were chosen because this is the starting point for obtaining stable solitons [8]. Then, a single step jump was made to a point in the parameter space \(\left(\Delta,\left|S\right|^{2}\right)\) and maintained for 1.5 \(\upmu\)s (\(\approx 1310\) \(t_{\text{ph}}\)). We chose a configuration \((a,b)\) that shifted the single soliton region to be accessible from the blue side of the comb generation region in the parameter space \(\left(\Delta,\left|S\right|^{2}\right)\). These values are \(a=2\pi\times 9.71\) [GHz] and \(b=15.48\), in which we observed the shift of DKS region taking into account that the should potentially be replicated in experiments. To emphasize, generating a mode interaction strength greater than \(a=2\pi\times 10\) [GHz] could be physically unfeasible in chip design because achieving this interaction with the geometric designs proposed in the literature poses a challenge [1; 34]. Hence, exploring values below this threshold is warranted to align with experimental limitations. Figure 1c,f show the simulation results for a single MRR and DCM, respectively. In these simulations, we show the number of peaks in the time domain for each endpoint  
\((\Delta,|S|^{2})\) after 3 \(\upmu\)s. In the first simulation with a single ring (see Figure 1c), we emphasize the presence of the thin black DKS region on the right side of the comb generation zone.