## 1 Introduction  
Mid-infrared (MIR) supercontinuum generation (SCG) plays an essential role in molecular spectroscopy [1; 2; 3], medical diagnostics [4], infrared countermeasures [5], etc. When high-order dispersion is present, the dispersive wave (DW) on the longer wavelength side can be generated during the process of SCG if the phase-matching condition is satisfied [2]. The DW can maintain the coherence of the pump source and help extend the spectral range significantly into the MIR regime [6]. Due to the rich applications in the MIR regime, high-performance spectrum generation in this regime has attracted significant interest. To evaluate the performance of the MIR SCG and DW, the conversion efficiency, bandwidth, and flatness of the spectrum are the crucial metrics, especially the spectrum covering 3-5 um band, which is one of the atmospheric transparency windows [1]. High efficiency is an important research direction. [PERSON] et al. designed a large cross-section silicon nitride (Si\({}_{3}\)N\({}_{4}\)) channel waveguide and engineered the dispersion. By pumping the structure at around 2 um, conversion efficiency of the MIR DW up to 35% was obtained [7]. Similar conversion efficiency of 36.26% was achieved on the LNOI platform when pumpingat around 2 \(\upmu\)m [8]. Broadband and flattened SCG and DW are preferred in practical applications because they enable parallel operation, improve the accuracy of the measurement, and reduce the degree of the difficulty in power equalization [9]. For example, in trace gas detection, a broadband and flattened supercontinuum and DW source without many cluttered dips can maintain a good spectral uniformity and distinguish the absorption spectrum of a detected gas molecular accurately, thus improving the robustness of the gas detection system. However, there is typically a tradeoff between a large bandwidth and high degree of flatness in conventional SCG. A promising approach for MIR SCG and DW generation is to use cheap and commercially matured mode-locked femtosecond fiber lasers at 1550 nm as the pump seed. For example, [PERSON] et al. leveraged such a fiber-to-chip system to pump Si\({}_{3}\)N\({}_{4}\) waveguides for MIR-DW generation [6]. The dispersion engineering of the Si\({}_{3}\)N\({}_{4}\) platform relies on the thick films fabricated with the photonic Damascene process [10] to compensate for the material dispersion, which is a non-commercial and complicated technique. For a single Si\({}_{3}\)N\({}_{4}\) waveguide structure, it is difficult to obtain a broadband and flattened DW at a longer wavelength phase-matched point. The conversion efficiency is limited to 1% [6]. In order to acquire a large bandwidth and a high flatness of the SCG and DW, [PERSON] et al. proposed parallel dual-coupled Si\({}_{3}\)N\({}_{4}\) waveguide with different widths [11]. Based on this structure, they produce a four zero-dispersion-wavelength (ZDW) dispersion profile. The flattened MIR-DW generation spanning from 2.7 to 3.7 \(\upmu\)m has been achieved, but the conversion efficiency is still limited to 1-5% [11] because of the primary-stage dispersion engineering. Although a four-ZDW dispersion profile can also be produced by a vertical stacked slot waveguide structure [12; 13; 14], it is difficult to grow or deposit the different materials layer by layer. Besides the Si\({}_{3}\)N\({}_{4}\) platform, lithium niobate (LN) is a competitive platform for wavelength conversion due to its strong \(\chi^{(2)}\) and \(\chi^{(3)}\) nonlinearities. For instance, SCG [15] and MIR DW generation [16] have been demonstrated in the bulk LN crystal. However, the LN crystal is bulky, thus presenting a limited dispersion engineering ability and high power consumption (sub-mj level). Thin-film lithium-niobate-on-insulator (LNOI) is an emerging branch of LN, which presents a nano-scale size and high-index contrast compared with the bulk LN crystal.