## 3 Results and Discussion
We note that the evolution dynamic after the soliton fission length is not accurate because the stimulated Raman scattering is neglected in this work. For the case that the evolution dynamic after the soliton fission length is focused, the stimulated Raman scattering effectof LN [15, 31] should be considered for more accurate results. We can see from Figure 3a that the spectrum initially broadens symmetrically around the pump wavelength in the frequency domain due to the self-phase modulation (SPM). Accordingly, in the time domain, the soliton goes through a temporal compression process as shown in Figure 3d. When the propagation length reaches the soliton fission length, i.e., \(l_{\text{fission}}=l_{\text{D}}/N_{\text{soliton}}\), around 0.83 mm, the spectrum starts to explode into a fairly wide range. Here, \(N_{\text{soliton}}=(l_{\text{D}}/l_{\text{NL}})^{1/2}\) is the soliton number, and \(l_{\text{D}}=\tau_{0}^{2}/\mid\beta_{2}\mid\) and \(l_{\text{NL}}=1/\gamma P_{0}\) are the dispersion and nonlinear lengths, respectively. Almost at the same time of soliton fission, the MIR-DW at around 2.5 \(\upmu\)m and NIR-DW at around 0.68 \(\upmu\)m are generated. As a result of the upward trend towards zero in the \(\beta_{\text{int}}\) curve around 2.5 \(\upmu\)m, the MIR-DW at this band is obtained. Correspondingly, in the time domain, Figure 3d shows the fundamental sub-solitons and the DW pulses are observed temporally emitted from the compressed high order soliton due to their different group velocity. As both the MIR-DW and NIR-DW are generated in the normal dispersion region, the pulse of MIR-DW on the longer wavelength side exhibits acceleration relative to the sub-solitons, while the pulse of NIR-DW at shorter wavelength side shows a delay. When the propagation length reaches around 0.9 mm, the power of the MIR-DW increases and the spectrum is flattened. Figure 3b shows the spectrum at \(L=0.9\) mm extracted from the spectral evolution profile in Figure 3a. We find that the spectrum is very smooth with a high spectral flatness of 6.31 dB achieved in the wavelength range of 1.92-3.55 \(\upmu\)m, calculated by Equation (8):  
\[F=\text{max}(\log_{10}(spectrum(\lambda_{1}))-\log_{10}(spectrum(\lambda_{2}))) \tag{8}\]  
where \(F\) is the spectral flatness, \(spectrum\) is the normalized output spectrum, and \(\lambda_{1}\) and \(\lambda_{2}\) are the wavelengths in a certain wavelength region. The conversion efficiency of the MIR-DW is as high as 18.48%. This is calculated by the ratio between the energy of the MIR-DW and the soliton pump, which corresponds to on-chip energy in experiments. However, the intensity of the MIR-DW at the phase-matching point of 3.63 \(\upmu\)m is weak due to the insufficient nonlinear accumulation at this short propagation length. When \(L=0.95\) mm, there are dual hump-shaped MIR-DWs in the output spectrum in Figure 3c. One is centered at around 2.46 \(\upmu\)m, and the other is located at about 3.65 \(\upmu\)m, which corresponds to the normal dispersion region in Figure 2e. Since the intensity of the MIR-DW at around 3.65 \(\upmu\)m is accumulated to a relatively high level, these dual hump-shaped MIR-DWs are obtained at this propagation length. The conversion efficiency of the MIR-DW is calculated as 19.31%. Figure 3e,f show that the normalized temporal pulse envelopes at the dual-coupled waveguide output exhibit the soliton fission effect which splits into two components. We further studied the influence on the MIR-DW generation when the pump condition changes, i.e., the pulse width \(\tau_{0}\) and the coupled pulse energy. We can see the spectral and temporal evolution with the propagation length in Figure 4a,d when the full-width at half maxima \(\tau_{\text{FWHM}}\) is reduced to 50 fs. As the peak power \(P_{0}\) is kept at 4.5 kW, the pulse energy is reduced to around 130 pJ. Figure 4a shows the similar soliton evolution process as mentioned above. However, the soliton fission length is different from the one with \(\tau_{\text{FWHM}}=75\) fs due to the reduced pulse width.