The multi-core silicon-rich nitride photonic integrated waveguide is presented here to numerically model the spanning of supercontinuum generation in the mid-IR region beyond the 4 μm wavelength range by engineering the dispersion profile such that maximum phase-matching can be achieved. Tunable dispersive wave generation has been observed in a long mid-IR region up to 5.2 μm wavelength by the variation of gap spacing between the cores. This unique geometry enables a higher phase matching from 1.2 μm to 2.6 μm wavelength range resulting in the flat dispersion profile in the near-IR region by coupling the fundamental mode in narrow waveguide core to the higher-order mode in wide waveguide core that gives rise to anomalous GVD which plays an essential role in compensating chromatic dispersion in the mid-IR region.