## 2 Characterization of Pump Power and Detuning Parameter Space  
To understand the influence of mode interactions in the DCM, we explored the operating regions in the parameter space \((\Delta,\left|S\right|^{2})\). This study is necessary to identify the region where DKSs are generated. The characterization of the operating regions depends on applying the Lugiato-Lefever equation (LLE) [8; 15]. The LLE serves as a mathematical framework to simulate and characterize the propagation of light within a microring resonator, modeled through the parameters of loss, coupling factor, and chromatic dispersion. This framework is decisive when responding to input pump and damping parameter variations, providing a comprehensive tool for our analysis. The DCM was modeled by incorporating an avoided mode crossing (AMX) in the LLE [17]. This AMX accounts for the optical effects caused by an auxiliary ring with a smaller radius within the main ring, as illustrated in Figure 1. The AMX on the physical device can be controlled using a microheater positioned on top of the auxiliary ring. The observed two peaks in the DCM (see Figure 1e) relative to the MRR result from mode interactions caused by AMX, where the resonance modes of the primary microring and the auxiliary ring couple strongly, manifesting as peaks in the spectral response. These peaks represent resonant states with altered mode density and enhanced light confinement, enabling specific dispersion profiles that support DKS generation. The LLE equation can be expressed as follows:  
\[t_{R}\frac{\partial E(t,\tau)}{\partial t}=\left[-\alpha-i\delta_{0}-iL\frac{ \beta_{2}}{2}\frac{\partial^{2}}{\partial\tau^{2}}+i\gamma L|E|^{2}\right]E+ \sqrt{\phi}E_{\text{in}}+i\sum_{\mu}\frac{a/2}{\mu-b}\tilde{E}_{\mu}(t)e^{i\mu\tau} \tag{1}\]  
where \(E(t,\tau)\) is the complex envelope of the total intracavity field, \(t\) is the time variable, \(\tau\) is the fast time variable, \(t_{R}\) is the round-trip time, \(\alpha\) is half of the total loss per round trip, \(\delta_{0}\) is the frequency detuning, \(L\) is the cavity length, \(\beta_{2}\) is the second-order dispersion coefficient, \(\gamma\) is the Kerr coefficient, \(\phi\) is the coupling coefficient between the waveguide and the microring resonator, \(E_{\text{in}}\) is the pump field, \(a\) and \(b\) are the strength and frequency of the mode interaction, respectively, and \(\tilde{E}_{\mu}(t)\) is the Fourier transform, defined as  
\[\tilde{E}_{\mu}(t)=\frac{1}{T_{R}}\int_{0}^{T_{R}}E(t,\tau)e^{-i\mu\tau}d\tau. \tag{2}\]  
The LLE equation incorporates AMX effects with the term  
\[i\frac{1}{\alpha}\sum_{\mu}\left[\frac{\frac{2}{\mu}}{\mu-b}\,\widetilde{\psi} _{\mu}(T)\,e^{i\mu\mu}\right] \tag{3}\]  
The parameter values for \(a\) and \(b\) were selected based on previous studies that validate this approach [16; 30; 31]. These studies highlight the role of AMX in facilitating DKS generation by lowering the threshold for soliton formation under certain dispersion conditions [30; 31]. This incorporation of AMX into the LLE framework allows us to simu  
Figure 1: (**a**) Microring resonator structure. (**b**) Dispersion profile of a single microring in the anomalous regime. (**c**) Npeak \((\Delta_{i},|S|^{2})\). (**d**) The structure of a two-coupled-microring resonator, with heater. (**e**) Dispersion with mode interactions (**f**) Npeak \((\Delta_{i},|S|^{2})\) with a two-coupled-microring structure. The blue point in (**f**) represents the initial point of the simulation. late and predict the influence of pump power and dissipation variations on modal dynamics and DKS formation. This approach is essential for understanding how device parameters impact DKS generation, providing insights for optimizing design in specific applications. Equation (1) can be written in terms of three normalized parameters.